BANSAL CLASSES TARGET LIT JEE 2007 XI (PQRS) CALORIMETRY & HEAT TRANSFER C O N T E N T S KEYCONCEPT EXERCISE-I EXERCISE-II EXERCISE-III ANSWER KEY THERMAL EXPANSION Definition of Heat: Heat is a form of energy which is transferred between a system and its sur rounding as a result of temperature difference only. Thermal Expansion : Expansion due to increase in temperature. 1. Type of thermal expansion Coefficient of expansion (i) Linear (ii) Superficial (iii) Volume T . 1 A / a = Lim At>0 / 0 A t P = Lim 1 AA A t A 0 A t y = Lim 1 AV At>o v 0 A t For temperature change At change in length Al = l 0 a At Area AA= A^At volume AV = V0yAt (a) (b) 2. For isotropic solids otj = a 2 = a 3 = a (let) so P =2a and y = 3aFor anisotropic solids p = otj + a 2 and y = a , + a 2 + a 3 Here , a 2 and a 3 are coefficient of linear expansion in X, Y and Z d irections. Variation in density : With increase of temperature volume increases so density decreases and vice-versa. H d = Note (0 (ii) 3. (1 + yAt) For solids values of y are generally small so we can write d = d0 (1-yAt) (usi ng bimomial expansion) y for liquids are in order of 10~ 3 For water density increases from 0 to 4C so y is -ve (0 to 4 C) and for 4 C to high er temperature y is +ve. At 4 C density is maximum. Thermal Stress: Arod of length 10 is clamped between two fixed walls with dista nce 10. If temperature is changed by amount At then F (area assumed to be constant) stress : A A/ strain = I so, or Y = F/A F/ 0 F A///0 AA I AaAt F = Y A a A t / o/ c/. c/-.. u. :/. :: 4. If a is not constant (i) (a varies with distance) Let a = ax+b i Total expansion = Jexpansion of length dx = | ( ax + b)dxAt (ii) ( a varies with tempearture) Let a = f (T) dx " x 1 dx T 2 A/ _ j "a/ 0 dT Ti Caution: If a is in C then put Tj and T2 in C. similarly if a is in K then put Tj and T2 in K. CAL ORIMETR Y Quantity of heat transfered and specific heat The amount ofheat needed to incerase the temperature of 1 gmofwaterfrom 14.5Cto 15.5CatSTP is 1 calorie dQ = mcdT'h Q = m [ C dT (be careful about unit of temperature, use units according to the given units of C) Ti Heat transfer in phase change Q = rnL L = latent heat of substance in cal/ gm/ C or in Kcal/ kg/ C Li c e = 80 cal/ gm for ice L steam = 5 4 0 C a l / g m HEAT- TRANSFER (A) Conduction : Due to vibration and collision of medium particles. (i) Steady State : In this state heat absorption stops and temperatur e gradient throughout the rod dT becomes constant i.e. = constant. dx (ii) Before steady state : Temp of rod at any point changes Note: If specific heat of any substance is zero, it can be considered alwa ys in steady state. 1. Ohm's law for Thermal Conduction in Steady State : Let the two ends of rod of length 1 is maintained at temp Tj and T2 ( Tj > T2 ) dQ T i ~ T 2 I Thermal current = L D 1 K-XH T 1 / Where thermal resistance RT h = 1 1 2. Differential form of Ohm's Law dQ dT dT = KA = temperature gradient dT dx dx KA T-dT dx / o/ c/. c/-.. u. :/. :: (B) Convection: Heat transfer due to movement of medium particles. (Q Radiation: Every body radiates electromagnetic radiation of all possible wa velength at all temp>0 K. 1. Stefan's Law: Rate of heat emitted by a body at temp T K from per unit are a E = GT 4 J/sec/m 2 dQ Radiation power dl = P = o AT 4 watt If a body is placed in a surrounding of temperature TsdQ ^ = c A( T 4 - T s 4 ) valid only for black body heat from general body Emissmty or emmisive power e = ~ h e a t f r o m If temp of body falls by dT in time dt dT _ _ j4x dt ~ mS s (dT/dt=rate of cooling) Newton's law of cooling If temp difference of body with surrounding is small i.e. T = Ts dT 4 e Aa r r 3 / then, dt mS dT - T ( T - T ) so dt a ( T - T ) Average form of Newtons law of cooling If a body cools from T j to T2 in time 51 T s - T 2 _ K 5t mS T, +T, -T dt mS (used generally in objective questions) (for better results use this generally in subjective) 4. Wein's black body radiation At every temperature (>0K) a body radiates energy radiations of all wavelengths. According to Wein's displacement law if the wavelength corresponding to maximum energy is Xm then XmT = b where b = is a constant (Wein's constant) T=temperature of body T 3 >T 2 >T, ess / o/ c/. c/-.. u. :/. :: txttctst : Q. 1 An aluminium container of mass 100 gm contains 200 gm of ice at - 20C. Heat is added to the system at the rate of 100 cal/s. Find the temperature of the system after 4 mi nutes (specific heat of ice = 0.5 and L = 80 cal/gm, specific heat of A1 = 0.2 cal/gm/C) Q. 2 A U-tube filled with a liquid of volumetric coefficient of 10 _ 5 /C lies in a vertical plane. The height of liquid column in the left vertical limb is 100 cm. The liquid in the left vertic al limb is maintained at a temperature = 0C while the liquid in the right limb is maintained at a temperatur e = 100C. Find thedifference in levels in the two limbs. Q.3 A thin walled metal tank of surface area 5m 2 is filled with water tank and contains an immersion heater dissipating 1 kW. The tank is covered with 4 cm thick layer of in sulation whose thermal conductivity is 0.2 W/m/K. The outer face of the insulation is 25C. Find th e temperature of the tank in the steady state Q.4 A glass flask contains some mercury at room temperature. It is found that at different temperatures the volume of air inside the flask remains the same. If the volume of mercury in the flask is 300 cm 3 , then find volume of the flask (given that coefficient of volume expansion o f mercury and coefficient oflinear expansion of glass are 1.8 x 10^(C) _ 1 and9x 10~ 6 (C) _ 1 respectively) Q.5 A clock pendulum made of invar has a period of 0.5 sec at 20C. If th e clock is used in a climate where average temperature is 30C, aporoximately. How much fast or slow wil l the clock run in 10 6 sec. ( ai l w a r =l xl O - 6 / C) Q.6 A pan filled with hot food cools from 50.1 C to 49.9 C in 5 sec. H ow long will it take to cool from 40.1 C to 39.9C if room temperature is 30C? Q.7 A composite rod made of three rods of equal length and cross-sect ion as shown in the fig. The thermal conductivities of the materials of the rods are K/2, 5K and K respec tively. The end A and end B are at constant temperatures. All heat entering the face A goes out of the end B there being no loss of heat from the sides of the bar. Find the effective thermal cond uctivity of the bar A B I I 1 1 K/2 5K K Q.8 An iron bar (Young's modulus = 10 1 1 N/ m 2 , a = 10" 6 /C) 1 m long and 10~ 3 m 2 in area is heated from 0C to 100C without being allowed to bend or expand. Find the compressive for ce developedinside the bar. Q.9 A solid copper cube and sphere, both of same mass & emissivity are heated to same initial temperature and kept under identical conditions. What is the ratio of their initial rate of fall of temperature? Q. 10 A cylindrical rod with one end in a stream chamber and other end in ice cause melting of 0.1 gm of ice/sec. If the rod is replaced with another rod of half the length and dou ble the radius of first and thermal conductivity of second rod is 1/4 that of first, find the rate of ice melting in gm/sec / o/ c/. c/-.. u. :/. :: Q. l l Three aluminium rods of equal length form an equilateral triangle ABC. T aking O (mid point of rod BC) as the origin. Find the increase in Y-coordinate per unit cha nge in temperature of the centre of mass of the system. Assume the length of the each rod is 2m, and a d = 4 v3 x10" 6 /C Q.12 Q. 13 Q.14 Q.15 Three conducting rods of same material and cross-section are shown in figu re. Temperature of A, D and C are maintained at 20C, 90C and 0C. Find the ratio of length BD and BC if there is no heat flow in AB 20C 90'C 0C If two rods of layer L and 2 L having coefficients of linear expansion a and 2a respectively are connected so that total length becomes 3 L, determine the average coefficie nt of linear expansion of the composite rod. A volume of 120 ml of drink (half alcohol + half water by mass) originally at a temperature of 25C is cooled by adding 20 gm ice at 0C. If all the ice melts, find the f inal temperature of the drink, (density of drink = 0.833 gm/cc, specific heat of alcohol = 0.6 cal/gm/C) A solid receives heat by radiation over its surface at the rate of 4 kW. The heat convection rate from the surface of solid to the surrounding is 5.2 kW, and heat is gene rated at a rate of 1.7 kW over the volume of the solid. The rate of change of the average tem perature of the solid is 0. 5 o Cs - 1 . Find the heat capacity of the solid. Q.16 The figure shows the face and interface temperature of a composi te slab containing of four layers of two materials having identical thickness. Under steady state condition, find the value of temperature 6. 20C 10C E -5C -10C2k 2k k = thermal conductivity Q.17 Q.18 Q.19 Two identical calorimeter A and B contain equal quantity of water at 20C. A 5 gm piece of metal X of specific heat 0.2 cal g 4 (C) _ 1 is dropped into A and a 5 gm piece of metal Y into B. The equilibrium temperature in A is 22C and in B 23C. The initial temperature of both the metals is 40C. Find the specific heat of metal Y in cal g" 1 (C)~ l Two spheres of same radius R have their densities in the ration 8 . 1 and the ratio of their specific heats are 1 : 4. If by radiation their rates of fall of temperature are same , then find the ratio of their rates of losing heat. In the square frame of side I of metallic rods, the corners A and C are maintained at Tj and T2 respectively. The rate of heat flow from A to Ci sa. IfA and D are instead maintained Tj & T2 respectivley find, find the total rate of heat flow. Q.20 A hot liquid contained in a container of negligible heat capacity l oses temperature at rate 3 K/min, just before it begins to solidify. The temperature remains constant fo r 30 min, Find the ratio of specific heat capacity of liquid to specific latent heat of fusion is in Kr 1 (given that rate of losing heat is constant). / o/ c/. c/-.. u. :/. :: Q. 21 A thermostatted chamber at small height h above earth's surface maintaine d at 30C has a clock fitted in it with an uncompensated pendulum. The clock designer correctly designs it for heig ht h, but for temperature of 20C. If this chamber is taken to earth's surface, the clock in it would click cor rect time. Find the coefficient of linear expansion of material of pendulum, (earth's radius is R) Q.22 The coefficient of volume expansion of mercury is 20 times the coef ficient of linear expansion of glass. Find the volume of mercury that must be poured into a glass vessel of vol ume V so that the volume above mercury may remain constant at all temperature. Q. 23 Two 50 gm ice cubes are dropped into 250 gm ofwater ion a glass. If the w ater was initially at a temperature of 25C and the temperature of ice -15C. Find the final temperature of water, (specific heat of ice = 0.5 cal/gm/C and L = 80 cal/gm) Q.24 Water is heated from 10C to 90C in a residential hot water heater at a rat e of 70 litre per minute. Natural gas with a density of 1.2 kg/m 3is used in the heater, which has a transfer efficiency of 32%. Find the gas consumption rate in cubic meters per hour, (heat combustion for natural gas is 8400 kcal/kg) Q.25 A metal rod A of 25cm lengths expands by 0.050cm. When its temperature is raised from 0C to 100C. Another rod B of a different metal of length 40cm expands by 0.040 cm for t he same rise in temperature. A third rod C of 50cm length is made up of pieces of rods A and B placed end to end expands by 0.03 cm on heating from 0C to 50C. Find the lengths of each portion of the composite rod. Q.26 A substance is in the solid form at 0C. The amount of heat added to this substance and its temperature are plotted in the following graph. If the relative specific heat capacity of the solid substance is 0.5, find from the graph (i) the mass of the substance; (ii) the specific latent heat of the melting process, and (iii) the specific heat of the substance in the liquid state. Q. 27 One end of copper rod of uniform cross-section and of length 1.5 meters is in contact with melting ice and the other end with boiling water. At what point along its length should a te mperature of200C be maintained, so that in steady state, the mass of ice melting is equal to that of steam produced in the same interval of time? Assume that the whole system is insulated from the surrounding s. Q.28 Two solids spheres are heated to the same temperature and allowe d to cool under identical conditions. Compare: (i) initial rates of fall of temperature, and (ii) initial rates of loss of heat. Assume that all the surfaces have the same emissivity and ratios of their radii of, specific heats and densities are respectively 1 : a, 1 : p, 1 : y. Q.29 A vessel containing 100 gm water at 0C is suspended in the middle of a room. In 15 minutes the temperature of the water rises by 2C. When an equal amount of ice is placed in th e vessel, it melts in 10 hours. Calculate the specific heat of fusion ofice. Q. 3 0 The maximum in the energy distribution spectrum of the sun is at 4753 A and its temperature is 6050K. What will be the temperature of the star whose energy distribution shows a maxi mum at 9506 A. / o/ c/. c/-.. u. :/. :: txttctsttt Q. 1 A copper calorimeter of mass 100 gm contains 200 gm of a mixtur e of ice and water. Steam at 100C under normal pressure is passed into the calorimeter and the temperature of the mixture is allowed to rise to 50C. If the mass of the calorimeter and its contents is now 330 gm, what was the ratio of ice and water in the beginning? Neglect heat losses. Given : Specific heat capacity of copper = 0.42 x 10 3 J kg _ 1 K"x , Specific heat capacity of water = 4.2 x 10 3 J kg^Kr 1 , Specific heat of fusion of ice = 3.36 x 10 5 J kg - 1 Latent heat of condensation of steam = 22.5 x 1Q 5 Jkg" 1 Q. 2 A n i soscet es triangte is form ed w ith a rod of length lx and coefficient of linear expansion OTJ for the base and two thin rods each of length l2 and coefficient of linear expans ion a 2 for the two pieces, if the distance between the apex and the midpoint of the base remain unchanged a s the temperatures /, varied show that 7 l 2 Q.3 A solid substance of mass 10 gm at - 10C was heated to - 2C (still in the solid state). The heat required was 64 calories. Another 880 calories was required to raise the tempera ture of the substance (now in the liquid state) to 1C, while 900 calories was required to raise the temperature from -2C to 3C. Calculate the specific heat capacities of the substances in the solid and liquid state in calories per kilogram per kelvin. Show that the latent heat of fusion L is related to the melting point temperature t m by L = 85400 + 200 t m . Q.4 A steel drill making 180 rpm is used to drill a hole in a block of s teel. The mass of the steel block and the drill is 180 gm. If the entire mechanical work is used up in prod ucing heat and the rate of raise in temperature of the block and the drill is 0.5 C/s. Find (a) the rate of working of the drill in watts, and (b) the torque required to drive the drill. Specific heat of steel = 0.1 and J = 4.2 J/cal. Use ;P = i o Q. 5 A brass rod of mass m = 4.25 kg and a cross sectional area 5 cm 2 increases its length by 0.3 mm upon heating from 0C. What amount of heat is spent for heating the rod? The coefficien t of linear expansic 1 for brass is 2xl 0 - 5 / K, its specific heat is 0.39 kJ/kg.K and the density of brass is 8.5 x 10 3 kg/m 3 . Q.6 A submarine made of steel weighing 10 9 g has to take 10 8g of water in order to submerge when the temperature of the sea is 10C. How much less water it will have to take in when the sea is at 15C? (Coefficient of cubic expansion of sea water = 2 x 10"VC, coefficient o f linear expansion of steel = 1.2 x 105 /C) Q. 7 A flow calorimeter is used to measure the specific heat of a liquid. Heat is added at a known rate to a stream of the liquid as it passes through the calorimeter at a known rate . Then a measurement of the resulting temperature difference between the inflow and the out flow points of the liquid stream enables us to compute the specific heat of the liquid. A liquid of density 0.2 g/cm 3 flows through a calorimeter at the rate of 10 cm 3 /s. Heat is added by means of a 250-W electric heating coil, and a temperature difference of 25 C is established in steady-state conditions between the inflow and the outflow points. Find the specific heat of the liquid. / o/ c/. c/-.. u. :/. :: Q.8 Toluene liquid of volume 300 cm 3 at 0C is contained in a beaker an another quantity of toluene of volume 110 cm 3 at 100C is in another beaker. (The combined volume is 410 cm 3 ). Determine the total volume of the mixture of the toluene liquids when they are mixed together. Given the coefficient of volume expansion y = 0.001/C and all forms of heat losses can be igno red. Also find the final temperature of the mixture. Q. 9 Ice at -20C is filled upto height h = 10 cm in a uniform cylindrical vesse l. Water at temperature 9C is filled in another identical vessel upto the same height h= 10 cm. Now, water from second vessel is poured into first vessel and it is found that level of upper surface falls t hrough Ah = 0. 5 cm when thermal equilibrium is reached. Neglecting thermal capacity o f vessels, change in density of water due to change in temperature and loss of heat due to ra diation, calculate initial temperature 0 of water. Given, Density of water, pw = 1 gm cm - 3 Density of ice, p. =0. 9gm/ cm 3 Specific heat of water, sw = 1 cal/gm C Specific heat of ice, s; = 0.5 cal/gmC Specific latent heat of ice, L = 80 cal/gm Q. 10 A composite body consists of two rectangular plates of the same dimens ions but different thermal conductivities KA and Kg. This body is used to transfer heat between t wo objects maintained atdifferent temperatures. The composite body can be placed such that flow of heat takes place either parallel to the interface or perpendicular to it. Calculate the effective therma l conductivities K. and Kj Of the composite body for the parallel and perpendicular orientations . Which orientation will have more thermal conductivity? Q. 11 Two identical thermally insulated vessels, each containing n mole of an ideal monatomic gas, are interconnected by a rod of length I and cross-sectional area A. Materi al of the rod has thermal conductivity K and its lateral surface is thermally insulated. If, at initial m oment (t = 0), temperature of gas in two vessels is T, and T2 (< T} ), neglecting thermal capacity of the r od, calculate difference between temperature of gas in two vessels as a function of time. Q. 12 A highly conducting solid cylinder of radius a and length I is su rrounded by a co-axial layer of a material having thermal conductivity K and negligible heat capacity. Temperat ure of surrounding space (out side the layer) is T0 , which is higher than temperature of the cylinder. If heat capacity per unit volume of cylinder material is s and outer radius of the layer is b, calculate time required to increase temperature of the cylinder from Tt to T r Assume end faces to b e thermally insulated. Q. 13 A vertical brick duct(tube) is filled with cast iron. The lower end of the duct is maintained at a temperature T, which is greater than the melting point Tm of cast iron and the upper end at a temperature T2 which is less than the temperature of the melting point of cast iron. It is g iven that the conductivity of liquid cast iron is equal to k times the conductivity of solid cast iron. Deter mine the fraction of the duct filled with molten metal. Q.14 Water is filled in a non-conducting cylindrical vessel of uniform cross-sectional area. Height of water column is h0 and temperature is 0C. If the vessel is exposed to an atmosph ere having constant temperature of - 0C (< 0C) at t = 0, calculate total height h of the column at ti me t .Assume thermal conductivity ofice to be equal to K.Density ofwater is pf f i and that of ice is p.. Latent heat of fusion of ice isL. / o/ c/. c/-.. u. :/. :: Q.15 A lagged stick of cross section area 1 cm 2 and length 1 m is initially at a temperature of 0C. It is then kept between 2 reservoirs of tempeature 100C and 0C. Specific heat capacity is 10 J/kgC and linear mass density is 2 kg/m. Find 100C oc (a) temperature gradient along the rod in steady state. (b) total heat absorbed by the rod to reach steady state. Q.16 A cylindrical block of length 0.4 m an area of cross-section 0.04 m 2 is placed coaxially on a thin metal disc of mass 0.4 kg and of the same cross-section. The upper face of the c ylinder is maintainedat a constant temperature of 400K and the initial temperature of the disc is 300K. If the thermal conductivity of the material of the cylinder is 10 watt/m-K and the specific heat of the material of the disc in 600 J/kg-K, how long will it take for the temperature of the di sc to increase to 350K? Assume, for purposes of calculation, the thermal conductivity of the disc to be very high and the system to be thermally insulated except for the upper face of the cylinder . Q.17 A copper calorimeter of negligible thermal capacity is filled with a liqui d. The mass of the liquid equals 250 gm. A heating element of negligible thermal capacity is immersed in the liq uid. It is found that the temperature of the calorimeter and its contents rises from 25C to 30C in 5 minutes when a or rent of 20.5 ampere is passed through it at potential difference of 5 volts. The liquid is thrown off and the heater is again switched on. It is now found that the temperature of the calorimeter alone is constantly maintained at 32C when the current through the heater is 7A at the potential difference 6 volts . Calculate the specific heat capacity of the liquid. The temperature ofthe surroundings is 25C. Q.18 A solid copper sphere cools at the rate of 2.8C per minute, when its temper ature is 127C. Find the rate at which another solid copper sphere of twice the radius lose its temperatu re at 327C, if in both the cases, the room temperature is maintained at 27C. Q.19 A calorimeter contains 100 cm 3 of a liquid of density 0.88 g/cm 3 in which are immersed a thermometer and a small heating coil. The effective water equivalent of cal orimeter, thermometer and heater may be taken to be 13 gm. Current of 2 A is passed thro ugh the coil. The potential difference across the coil is 6.3 V and the ultimate steady state temperature i s 55C. The current is increased so that the temperature rises slightly above 55C, and then i t is switched off. The calorimeter and the content are found to cool at the rate of 3.6C/min. (a) Find the specific heat of the liquid. (b) The room temperature during the experiment was 10C. If the room te mperature rises to 26C, find the current required to keep the liquid at 55C. You may assume that Newt on's law is obeyed and the resistance of the heater remains constant. Q.20 End A of a rod AB of length L = 0.5 m and of uniform cross-sectional a rea is maintained at some constant temperature. The heat conductivity of the rod is k = 17 J/s-rnK. T he other end B of this rod is radiating energy into vacuum and the wavelength with maximum e nergy density emitted from this end is X Q = 75000 A. If the emissivity of the end B is e = 1, determine the temperature of the end A. Assuming that except the ends, the rod is thermally insulated .Q.21 A wire of length 1.0 m and radius 10" 3 m is carrying a heavy current and is assumed to radiate as a blackbody. At equilibrium temperature of wire is 900 K while that of the s urroundings is 300 K. The resistivity of the material of the wire at 300 K is n 2 x 10" 8 O-m and its temperature coefficient of resistance is 7.8 x 10' 3 /C. Find the current in the wire, [a = 5.68 x 10" 8 w/m 2 K 4 ]. / o/ c/. c/-.. u. :/. :: Q.22 The temperature distribution of solar radiation is more or less same as that of a black body whose maximum emission corresponds to the wavelength 0.483 jam. Find the rate of cha nge of mass due to radiation. [Radius of Sun = 7.0 x 10 8 m] Q.23 A black plane surface at a constant high temperature Th , is parallel to another black plane surface at constant lower temperature T; . Between the plates is vacuum. In order to red uce the heat flow due to radiation, a heat shield consisting of two thin black plates, thermally isolated from each other, it placed between the warm and the cold surfaces and parallel to these. After some time st ationary conditions are obtained. By what factor r) is the stationary heat flow reduced due to the prese nce of the heat shield? Neglect end effects due to the finite size of the surfaces. Q.24 The shell of a space station is a blackened sphere in which a tempera ture T = 500K is maintained due to operation of appliances of the station. Find the temperature o f the shell if the station is enveloped by a thin spherical black screen of nearly the same radius as the radius of the shell. Q.25 A liquid takes 5 minutes to cool from 80C to 50C. How much time will it take to cool from 60C to 30C ? The temperature of surrounding is 20C. Use exact method. Q .26 Find the temperature of equilibrium of a perfectly black disc exposed normally to the Sun's ray on the surface of Earth. Imagine that it has a nonconducting backing so that it can radiate only to hemisphere of space. Assume temperature of surface of Sun = 6200 K, radius of su n = 6.9 * 10 s m, distance between the Sun and the Earth = 1.5 x l o 1 1 m. Stefan's constant = 5.7 x i0~ s W/m2 .K 4 . What will be the temperature if both sides of the disc are radiate? Blackened envelop / o/ c/. c/-.. u. :/. :: txttctst ttt Q. 1 The temperature of 100 gm of water is to be raised from 24 C to 90 C by adding steam to it. Calculate the mass of the steam required for this purpose. [JEE '96] Q.2 Two metal cubes A & B of same size are arranged as shown in figure. The extreme ends of the combination are maintained at the indicated temperatures. The arrangement is thermally insulated. The coefficients of thermal conductivity of A & B are 300 W/mC and 200 W/mC respectively. After steady state is reached the temperature T of the interface will be . [JEE' 96] Q.3 A double pane window used for insulating a room thermally from ou tside consists of two glass sheets each of area 1 m 2 and thickness 0.01 m separated by a 0.05m thick stagnant air space. In the steady state, the room glass interface and the glass outdoor interface are at constant temperatures of 27C and 0C respectively. Calculate the rate of heat flow through the windo w pane. Also find the temperatures of other interfaces. Given thermal conductivities of g lass and air as 0.8 and 0.08 W nr' K1 respectively. [JEE'97] Q. 4 The apparatus shown in the figure consists of four glass columns connected by horizontal sections. The height of two central columns B & C are 49 cm each. The two outer columns A & D are open to the atmosphere. A & C are maintained at a temperature of 95 C while the columns B & D are maintained at 5 C. The height of the liquid in A & D measured from the base line are 52.8 cm & 51 cm respectively. Determine the coefficient of thermal expansion of the liquid, [JEE '97] Q.5 A spherical black body with a radius of 12 cm radiates 450 W power at 5 00 K . If the radius were halved and the temperature doubled, the power radiated in watt would be : (A) 225 (B) 450 (C) 900 (D) 1800 Q.6 Earth receives 1400 W/m 2 of solar power . If all the solar energy falling on a lens of area 0.2 m 2 is focussed on to a block of ice of mass 280 grams, the time taken to melt the ice will be minutes. (Latent heat of fusion of ice = 3.3 x 10 5 J/kg) [JEE '97] Q.7 A solid body X of heat capacity C is kept in an atmosphere whose temp erature is TA = 300K. At time t = 0, the temperature of X is T0 = 400K. It cools according to Ne wton's law of cooling. At time tj its temperature is found to be 3 5 OK. At this time t p the body X is connected to a larger body Y at atmospheric temperature TA , through a conducting rod of length L , cross-sectional area A and thermal conductivity K. The heat capacity of Y is so large that any variati on in its temperature may be neglected. The cross-sectional area A of the connecting rod is small compared to the surface area of X. Find the temperature of X at time t = 3t r [JEE' 98] Q.8 A black body is at a temperature of2880 K. The energy of radiation emitted by this obj ect with wavelength between 499 nm and 500 nm is Up between 999 nm and 1000 nm is U2 and between 1499 nm and 1500nmisU3 . TheWienconstantb = 2.88 x 10 6 nmK. Then [JEE' 98] (A) Uj = 0 (B)U3 = 0 (C) Uj > U2 ( D) U2 >U1 o A B A 95 C 95 / o/ c/. c/-.. u. :/. :: Q.9 A bimetallic strip is formed out of two identical strips one of copper and the other of brass. The coefficient of linear expansion of the two metals are a c and ctg. On heating, the temperat ure of the strip goes up by AT and the strip bends to form an arc of radius of curvature R. Then R is: (A) proportional at AT (B) inversely proportional to AT [JEE' 99] (C) proportional to lOg - a c | (D) inversely proportional to | aB - a c | Q.10 A block of ice at - 10C is slowiy heated and converted to steam at 100C. Which of the following curves represents the phenomenon qualitatively? [JEE (Scr) 2000] (A) (B) \ (C) Heat supplied Heat supplied Heat supplied (D) Heat supplied Q. 11 The plots of intensity versus wavelength for three black bodies a t temperature T, , T2 and T, respectively are as shown. Thentemperatures are such that ( A) T1 >T2 >T3 (C) T2 > T3 > T1 [JEE (Scr) 2000] (B) T j > T3 > T2 (C) T. > T2 > Tt Q . 1 2 Three rods made of the same material and having the same cros s-section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at 0C and 90C respectively. The temperature of the junction of the three rods will be [JEE(Scr)2001 ] (A) 45C (B) 60C (C) 30C (D)20C oc,S0C "90CQ. 13 An ideal black body at room temperature is thrown into a furnace. I t is observed that (A) initially it is the darkest body and at later times the brightest. (B) it the darkest body at all times (C) it cannot be distinguished at all times. (D) initially it is the darkest body and at later times it cannot be dis tinguished. [JEE(Scr)2002] Q. 14 An ice cube of mass 0.1 kg at 0C is placed in an isolated container which is at 227C. The specific heat S of the container varies with temperature T according the empir ical relations = A + BT, where A= 100 cal/kg-K and B = 2 x 10~ 2 cal/kg-K 2 . If the final temperature of the container is 27C, determine the mass of the container. (Latent heat of fusion for water = 8 x \ o 4 cal/kg. Specific heat of water = 10 3 cal/kg-K) [JEE' 2001] Q.15 Two rods one of aluminium of length /, having coefficient of linear expan sion a a , and other steel of length l 2 having coefficient of linear expansion a s are joined end to end. The expansion in both the h rods is same on variation of temperature. Then the value of , . r is n +/ 2 (A) a c a a + a s (B) a 0 a a - a s (C) Otc [JEE (Scr) 2003] (D) None of these / o/ c/. c/-.. u. :/. :: Q.16 Q.21 2 kg ice at - 20C is mixed with 5 kg water at 20C. Then final amount ofwater in t he mixture would be; Given specific heat of ice = 0.5cal/gC, specific heat ofwater = 1 cal/gC, Q.17 Q.18 (a) (b) Q.19 Latent heat of fusion of ice = 80 cal/g. (A) 6 kg (B) 5 kg (C) 4 kg If emissivity of bodies X and Y are ex and ey and absorptive powerare Ax and Ay then [JEF (Scr) 2003] (A) ey > e x ; Ay > Ax (B) e y < e x ; A y < A x ( C ) e y > e x ; Ay < Ax (D) ey = e x ; Ay = Ax Hot oil is circulated through an insulated container with a wooden lid at the top whose conductivity K = 0.149 J/(m-C-sec), thickness t = 5 mm, emissivity = 0.6. Temperature of the top of the lid in steady state i s at T, = 127. If the ambient temperature Ta = 27C. Calculate rate of heat loss per unit area due to radiation from the lid. 17 _8 temperature of the oil. (Given a = x 10 ) [JEE (Scr) 2003] (D) 2 kg V.T a =27C -= Hot oil [JEE 2003] Q.20 Three discs A, B, and C having radii 2 m, 4 m and 6 m respectively are coat ed with carbon black on their outer surfaces. The wavelengths corresponding to maximum intensity are 300 nm, 400 nm and 500 nm respectively. The power radiated by them are Q A , Q B and Q C respectively, (a) Qa is maximum (B) Q B is maximum [JEE' 2004 (Scr.)] (C) Q C is maximum (D) Q A = Q B = Q C Two identical conducting rods are first connected independently to two vessels, one containing water at 100C and the other containing ice at 0 C. In the second case, the ro ds are joined end to end and connected to the same vessels. Let qj and q2 g/s be the rate of melting of ice in the two cases respectively. The ratio q9 /qT is (A) 1/2 (B) 2/1 (C) 4/1 (D) 1/4 [JEE'2004 (Scr.)] Liquid oxygen at 50 K is heated to 300 K at constant pressure of 1 atm. The rate of heating is constant. Which of the following graphs represents the variation of temperat ure with time? Temp.f Temp.f , Temp.f Temp. (A) (B) Time (C)Time (D) Time Q.22 Q.23 Time [JEE' 2004 (Scr.)] A cube of coefficient of linear expansion a s is floating in a bath containing a liquid of coefficient of volume expansion y t When the temperature is raised by AT, the depth upto which the cub e is submerged in the liquid remains the same. Find the relation between a s and yb showing all the steps. [JEE 2004] One end of a rod of length L and cross-sectional area A is kept in a furnace of temperature T r The other end of the rod is kept at a temperature T2 . The thermal conductivity of the material of the rod is K and emissivity of the rod is e. It is given that T2 = Ts + AT where AT Ts , Ts being the temperature of the surroundings. If AT oc (Tj - Ts ), find the proportionality constant. Consider that heat is lost only by radiation at the end where the temperature of the rod is T2 . [JEE 2004] Furance T f Insulated Furance T f Rod Furance T f * L * Insulated / o/ c/. c/-.. u. :/. :: Q. 24 Three graphs marked as 1,2,3 representing the variation of maximum emissi ve power and wavelength of radiation of the sun, a welding arc and a tungsten filament. Which of the following combination is correct (A) 1-bulb, 2 > welding arc, 3 > sun (B) 2-bulb, 3 welding arc, 1 - sun (C) 3-bulb, 1 welding arc, 2 sun (D) 2-bulb, 1 -> welding arc, 3 sun [JEE' 2005 (Scr)] Q. 25 In which of the following phenomenon heat convection does not take place (A) land and sea breeze (B) boiling of water (C) heating of glass surface due to filament of the bulb (D) air around the furance [JEE' 2005 (Scr)] Q.26 2 litre water at 27C is heated by a 1 kW heater in an open container. On an average heat is lost to surroundings at the rate 160 J/s. The time required for the temperature to reach 77C is (A) 8 min 20 sec (B)10min (C)7min (D)14min [JEE' 2005 (Scr)] Q.27 A spherical body of area A and emissivity e = 0.6 is kept inside a black body. What is the rate at which energy is radiated per second at temperature T (A) 0.6 a AT 4 (B)0. 4aAT 4 ( C) 0. 8cAT4 (D)l . OaAT 4 [JEE 2005 (Scr)] Q. 28 1 calorie is the heat required to increased the temperature of 1 gm of water by 1 C from (A) 13.5Cto 14.5C at 76 mm of Hg (B) 14.5Cto 15. 5Cat 760mmofHg (C) 0C to 1C at 760 mm of Hg (D) 3C to 4C to 760 mm of Hg [JEE* 2005 (Scr)] / o/ c/. c/-.. u. :/. :: Q.i Q.5 25.5C 5 sec slow / 6 M/3 Q.2 Q.6 0.1 cm 10 sec ANSWER KEY txttctst : Q.3 65C Q.7 15K/16 Q.4 Q.8 2000 cm 3 10, 000 N Q.9 .71. Q.10 0.2 Q.12 7/2 Q.13 5 a/3 Q.16 5C Q.17 27/85 Q.20 1/90 Q.21 h/5R Q.24 104.2 Q.25 10cm, Q.27 10.34 cm Q.28 ctPy: Q. l l 4 x 10 - 6 m/ C Q.14 4C Q.15 1000 J (C 0 )1 Q.18 2 : 1 Q.22 3Y/ 20 Q.19 (4/3) Q.23 0 C : a 2 Q.29 80 k cal/kg txttctsttt Q.30 3025 K Q.I 1 : 1.26 Q.3 800 cal kg" 1 K 1 , 1000 cal kg" 1K - 1 Q.4 (a) 37.8 J/s (Watts), (b) 2.005 N-m Q.5 25 kJ Q.6 9.02 x 10 5 gm Q.7 5000 J/C kg Q.8 decrease by 0.75 cm 3 , 25C Q.9 45C T . K A + K R V 2 KA KB Q.10 K > Kj_, K| = ; K x a 2 s . Q.12 ^ l o g 1 B Q. l l (T, ~T2 ) e ( 4KAt N | "\3nRi J l0g e V. T 0 ~ T 2 J / \ 1 1 JBL 1 V Q.14 h0 + Q.17 21000 Jkg^Kr 1 Q.20 T a = 423 K Q.23 r| = 3 Q.25 10 minutes Q.I 12 gm Q.4 2 x 10^ C l oge 2 12k;6t Pi L f Q.2 60 C Q.5 D k(Tt - T m ) Q 1 3 I k ( T1 - Tm ) + ( T m - T 2 ) Q.15 (a) 100 C/m, (b) 1000 J Q.16 166.3 sec Q.18 9.72C/min Q.19 (a)0.42 cal/gmC, (b) 1.6A Q.21 36A Q.22 ~ = 5.06 x 10 9 kg/s dtQ.24 T" = x 500 = 600 K Q.26 T0 = 420 K, T0 = 353.6 K txttctstttt Q.3 41.53 Watt; 26.48 C;0.55C Q.6 5.5 min Q.7 k = ; T = 300 + 50 exp. [LC tj Q.8 D Q.9 B, D Q.10 A Q. l l B Q.12 B Q.13 D Q.15 A Q.16 A Q.17 A Q.18 (a) 595 watt/m 2 , ( b ) T0 * 4 2 0 K K Q.20 D Q.21 C Q.22 y,= 2a s Q.23 Q.25 C Q.26 A Q.27 A Q.28 B 4eaLTf +K Q.14 0.5 kg Q.19 B Q.24 A / o/ c/. c/-.. u. :/. :: BA TARGET IIT JEE 2007 XII (ALL) C O H T E N T S KEYCONCEPTS EXERCISE-1 EXERCISE-II EXERCISE-III ANSWER KEY KEY CONCEPTS 1. CAPACITANCE OF A N ISOLATED SPHERICAL CONDUCTOR : C = 471 e 0 e ( R in a medium C = 47C G R in air 3 . 4. This sphere is at infinite distance from all the conductors. The Capacitance C = 47T E Q R exists between the surface of the sphere & earth . SPHERICAL CAPACITOR : It consists of two concentric spherical shells as shown in figure. Here capacita nce of region between the two shells is Ct and that outside the shell is C2 . We have 471 en ab C = b - a and C2 = 471 e Q b Depending on connection, it may have different combinations of C, and -C2. PARALLEL PLATE CAPACITOR : ( i ) UNIFORM DI-ELECTRIC MEDIUM : If two parallel plates each of area A & separated by a distance d are charged with equal & opposite charge Q, then the system is called a parallel plate capacitor & its capacitance is given by, ^ S)6r A . C = ; in a medium C = with air as medium UThis result is only valid when the electric field between plates of capacitor is constant, ( i i ) MEDI UM PARTLY AI R : C = So A d - l t - i When a di-electric slab of thickness t & relative permittivity e r is introduced between the plates of an air capacitor, then the distance between the plates is effectively reduced by the di-electric slab . ( i i i ) COMPOSITE MEDI UM : V ^rJ c = irrespective of the position of l l l l P 3 BSSSSii P 3 BSSSSii G0 A I I - r l r2 r3 CYLINDRICAL CAPACITOR : It consist of two co-axial cylinders of radii a& b, the outer conductor is earth ed. The di-electric constant of the medium filled in the space between the cylinder is 2ne n e Farad e r . The capacitance per unit length is C = y r in m /.o/ c/. c+t+ct:+ct ::: CONCEPT o r VARIATION OF PARAMETERS: 6. 9. 10 As capacitance of a parallel plate capacitor isC = e 0 kA , if either of k, A or d varies in the region between the plates, we choose a small dc in between the plates and for total capacitance of system. If all dC's are in series 1 dx -, If all dC's are in parallel CT = } dC J e 0 k(x)A(x) COMBINATION OF CAPACITORS : ( i ) CAPACITORS IN SERIES : In this arrangement all the capacitors when uncharged get the same charge Q but the potential difference across each will differ (if the capacitance are unequal). Q Q Q rIMHh C | C2 C3 v, v, v,1 1 1 + 1 + + C3 + 1 (ii) CAPACITORS IN PARALLEL : When one plate of each capacitor is connected to the positive terminal of the battery & the other plate of each capacitor is connected to the negative terminals of the battery, then the capacitors are said to be in parallel connection. The capacitors have the same potential difference, V but the charge on each one is different (if the capacitors are unequal). % Cj.V s 1 c,,v % 1 jC 3 ,y 1 Q +v eq. C I + C 2 + C 3 + + c ENERGY STORED IN A CHARGED CAPACITOR : Capacitance C, charge Q & potential difference V; then energy stored is 1 1 1 Q 2 U = - CV 2 = QV = - . This energy is stored in the electrostatic field set up in the di-electric medium between the conducting plates of the capacitor . HEAT PRODUCED IN SWITCHING IN CAPACITIVE CIRCUIT Due to charge flow always some amount of heat is produced when a switch is close d in a circuit which can be obtained by energy conservation as Heat = Work done by battery - Energy absorbed by capacitor. SHARING OF CHARGES : When two charged conductors of capacitance Cs & C2 at potential V} & V2 respectively are connected by a conducting wire, the charge flows from higher potential conductor to lower potential conductor, until the potential of the two condensers becomes equal. The com mon potential (V) after sharing of charges; net charge _ q j + q 2 C, + C2 V = C,V1 + C2 V2 ( V, - V2 )2 net capacitance C, + C2 Ct + C2 charges after sharing qj = C,'V & q2 = C2V. In this process energy is lost in the connecting wire C C as heat. This loss of energy is Ui n i t i a l - Ue a l = ^ r ^ g REMEMBER : (i) The energy of a charged conductor resides outside the conductor in its EF , where as in a condenser it is stored within the condenser in its EF. (ii) The energy of an uncharged condenser = 0 . (iii) The capacitance of a capacitor depends only on its size & geometry & the di-electric between the conducting surface .(i.e. independent of the conductor, like, whether it is copp er, silver, gold etc) ,o/ c/. c+t+ct:+ct txttctst - t Q.i Q.2 Q.3 Q.4 Q.5 Q.6 .CO, A solid conducting sphere of radius 10 cm is enclosed by a thin metallic shell of radius 20 cm. A charge q = 20pC is given to the inner sphere. Find the heat generated in the p rocess, the inner sphere is connected to the shell by a conducting wire The capacitor each having capacitance C = 2pF are connected with a battery of emf 30 V as shown in figure. When the switch S is closed. Find (a) the amount of charge flown through the battery (b) the heat generated in the circuit (c) the energy supplied by the battery (d) the amount of charge flown through the switch S The plates of a parallel plate capacitor are given charges +4Q and -2Q. The capa citor is then connected across an uncharged capacitor of same capacitance as first one (= C). Find the f inal potential difference between the plates of the first capacitor. +i, In the given network if potential difference between p and q is 2V and C2 = 3Cr Then find the potential difference between a&b. ' 3 0 V H Mq C, C. Find the equivalent capacitance of the circuit between point A and B. c 2C 11 4C 11 8C 11 11 - C : 11 11 : C :11 ! 1 11 \ \ \ \ \ r c yInfinite / section/ c 11 2C 11 4C II 8C +3 q +q The two identical parallel plates are given charges as shown in figure. If the plate area of either face of each plate is A and separation between plates is d, then find the amount of heat liberate after closing the switch. Q. 7 Find heat produced in the circuit shown in figure on closing the switch S. Q.8 In the following circuit, the resultant capacitance between A and B is 1 pF. Find the value of C. T T 2 ^ f Q.9 Three capacitors of 2pF, 3pF and 5|iF are independently charged with batteries of emf' s 5V, 20V and 10V respectively. After disconnecting from the voltage sources. These capacitors are connected as shown in figure with their positive polarity plates are connected to A and negative polarity is earthed. Now a battery of 20V and an uncharged capacitor of 4jaF capacitance are connected to the junction A as shown with a switch S. When switch is closed, find : (a) the potential of the junction A. (b) final charges on all four capacitors. 2\xV ^Slr 5NF \ I' ^III 20V 4|j.F 4r /.o/ c/. c+t+ct:+ct ::: Q.10 Find the charge on the capacitor C = 1 pF in the circuit shown in the f igure. 7 Iph IjxK IpF IpF C-luF l(iF :pnF yUlF: :IMF : Q. l l Find the capacitance of the system shown in figure. k = 1 k = 2 k = 3 k = 4 Q.12 The figure shows a circuit consisting of four capacitors. Find the effective capacitance between X and Y.Q. 13 Five identical capacitor plates, each of area A, are arranged such that adjacent plates are at a distance'd* apart, the plates are connected to a source of emf V as shown in figure. The charge on plate 1 is and that on plate 4 is . V+ Q.14 In the circuit shown in the figure, intially SW is open. When the switch is closed, the charge passing through the switch in the direction to X AEI 60 V 2 nF 60 V SW 3 1 J Q.15 Q.16 In the circuit shown in figure, find the amount of heat generated when switch s is closed. Two parallel plate capacitors of capacitance C and 2C are connected in parallel then following steps are performed. (i) Abattery of voltage V is connected across points A and B. (ii) A dielectric slab of relative permittivity k is slowly inserted in capacito r C. (iii) Battery is disconnected. (iv) Dielectric slab is slowly removed from capacitor. Find the heat produced in (i) and work done by external agent in step (ii) & (iv ). Q.17 The plates of a parallel plate capacitor are separated by a distance d = 1 cm. Two parallel sided dielectric slabs of thickness 0.7 cm and 0.3 cm fill the space between the plates. If the d ielectric constants of the two slabs are 3 and 5 respectively and a potential difference of440V is applied acro ss the plates. Find: (i) the electric field intensities in each of the slab s. (ii) the ratio of electric energies stored in the first to that in the second d ielectric slab. Q.18 A 10 pF and 20 pF capacitor are connected to a 10 V cell in parallel fo r some time after which the capacitors are disconnected from the cell and reconnected at t = 0 with each oth er, in series, through wires of finite resistance. The +ve plate of the first capacitor is connected to the -ve plate of the second capacitor. Draw the graph which best describes the charge on the +ve plate of th e 20 pF capacitor with increasing time. List of recommended questions from LE. Irodov. 3.101, 3.102, 3.103, 3.113, 3.117, 3.121, 3.122, 3.123,3.124, 3.132,3.133, 3.1 41,3.142, 3.177,3.184, 3.188. 3.199. 3.200,3.201. 3.203, 3.204. 3.205 /.o/ c/. c+t+ct:+ct ::: txttctst - tt Q. 1 (a) For the given circuit. Find the potential difference across all the ca pacitors,(b) How should 5 capacitors, each of capacities, l pF be connected so as to produce a total capacitance of 3/7 pF. 6oF, Ih-H^f ' I h 9|iF 8(xF +. 25V Q.2 The gap between the plates of a plane capacitor is filled with an isotropic insulator whose di-electric 71 constant varies in the direction perpendicular to the plates according to the la w K=Kj 1 + sin X L d where d is the separation, between the plates & Kt is a constant. The area of t he plates is S. Determine the capacitance of the capacitor. Q.3 Five identical conducting plates 1,2,3,4 & 5 are fixed parallel 5 to and equdistant from each other (see figure). Plates 2 & 5 are connected by a conductor while 1 & 3 are joined by another conductor. The junction of 1 & 3 and the plate 4 are connected to a source of constant e.m.f. V0 . Find; (i) the effective capacity of the system between the terminals of the source. (ii) the charges on plates 3 & 5. Given d = distance between any 2 successive plates & A= area of either face of each plate . Q.4 Apotential difference of300 Vis applied between the plates of a plane capac itor spaced 1 cm apart. A plane parallel glass plate with a thickness of 0.5 cm and a plane parallel paraf fin plate with a thickness of 0.5 cm are placed in the space between the capacitor plates find : (j) Intensity of electric field in each layer. (ii) The drop of potential in each layer. (iii) The surface charge density of the charge on capacitor the plates. Given t hat: kg l a s s = 6, kp a r a f f i n = 2 Q.5 A charge 200pC is imparted to each of the two identical parallel plate cap acitors connected in parallel. At t =0, the plates of both the capacitors are 0.1 m apart. The plates of first capacitor move towards each other with relative velocity 0.001 m/s and plates of second capacitor move apart with the same velocity. Find the current in the circuit at the moment. Q.6 A parallel plate capacitor has plates with area A & separation d . A bat tery charges the plates to a potential difference of V0 . The battery is then disconnected & a di-electric sl ab of constant K& thickness d is introduced. Calculate the positive work done by the system (capacitor + sl ab) on the man who introduces the slab. Q.7 A capacitor of capacitance C0 is charged to a potential V0 and then isolat ed. A small capacitor C is then charged from C0 , discharged & charged again, the process being repeated n times . The potential of the large capacitor has now fallen to V. Find the capacitance of the small capa citor. If V0 = 100 volt, V=35volt, find the value of n for C0 = 0.2 pF & C = 0.01075 pF . Is it possi ble to remove charge on C0 this way? Q. 8 When the switch S in the figure is thrown to the left, the plates of capac itors . VC, acquire a potential difference V. Initially the capacitors C2 C3 ar e uncharged. Thw switchis now thrown to the right. What are the final charges q p q2 & q3 on the corresponding capacitors. TLPI I c T /.o/ c/. c+t+ct:+ct ::: Q.9 A parallel plate capacitor with air as a dielectric is arranged horizontall y. The lower plate is fixed and the other connected with a vertical spring. The area of each plate is A. In the steady position, the distance between the plates is d0 . When the capacitor is c onnected with an electric source with the voltage V, a new equilibrium appears, with the distance between the plates as d r Mass of the upper plates is m. (1) Find the spring constant K. (ii) What is the maximum voltage for a given K in which an equilibrium is possi ble ? (lii) What is the angular frequency of the oscillating system around the equili brium value dj. (take amplitude of oscillation d{ ) Q.10 An insolated conductor initially free from charge is charged by repeated c ontacts with a plate which after each contact has a charge Q due to some mechanism. If q is the charge on the con ductor after the first operation, prove that the maximum charge which can be given to the conductor in this way is ~ Qq Q. l l A parallel plate capacitor is filled by a di-electric whose relative per mittivity varies with the applied voltage according to the law = aV, where a = 1 per volt. The same (but cont aining no di-electric) capacitor charged to a voltage V = 156 volt is connected in parallel to the fir st "non-linear" uncharged capacitor. Determine the final voltage Vf across the capacitors. Q.12 A capacitor consists of two air spaced concentric cylinders. The outer of radius b is fixed, and the inner is of radius a If breakdown of air occurs at field strengths greater than E^, show th at the inner cylinder should have (i) radius a = b/e if the potential of the inner cylinder is to be maximum (ii) radius a = b/Ve if the energy per unit length of the system is to be max imum. ,.JT 5 V - r Q. 13 Find the charge flown through the switch from Ato B when it is closed. 6 m F Jr~ Q.15 4=6nF 5V :d=6nf Q.14 Figure shows three concentric conducting spherical shells with inner and outer shells earthed and the middle shell is given a charge q. Find the electrostatic energy of the system stored in the region I and II. The capacitors shown in figure has been charged to a potential difference of V volts, so that it carries a charge CV with both the switches Sj and S2 remaining open. Switch Sj is closed at t=0. At t=R,C switch Sj is opened and S2 is closed. Find the charge on the capacitor at t=2RjC + R^C. His, s, Q.16 In the figure shown initially switch is open for a long time. Now the switch is closed at t = 0. Find the charge on the rightmost capacitor as "yv a function of time given that it was intially unchanged. /.o/ c/. c+t+ct:+ct ::: Q.17 In the given circuit, the switch is closed in the position 1 at t = 0 and then moved , I V to 2 after 250 p,s. Derive an expression for current as a function of time for J^ov [ 2 t > 0. Also plot the variation of current with time. I X 40V Q.18 Find the charge which flows from point Ato B, when switch is closed. txttctst - ttt VL 5(IF 5NF 5^F 5(.IF 5(IF 120V :500FJ :0.5 NF A B 10V Q. 1 Two parallel plate capacitors A&B have the same separation d=8.85 x lO^m between the plates. The plate areas of A & B are 0.04 m 2 & 0.02 m 2 respectively. A slab of di-electric constant (relative permittivity) K=9 has dimensions such that it can exacdy fill the space between the plates of capacitor B. (i) the di-electric slab is placed inside A as shown in the figure (i) Ais then charged to a potential difference of 110 volt. Calculate the capacitance of A and the energy stored in it. (ii) the battery is disconnected & then the di-electric slab is removed from A. Find the work done by the external agency in removing the slab from A. (iii) the same di-electric slab is now placed inside B, filling it complet ely. The two capacitors A& B are then connected as shown in figure (iii). Calculate the energy stored in the system. [ JEE ' 93, 7] Q.2 Q.3 Q.4 Q.5 Two square metallic plates of 1 m side are kept 0.01 m apart, like a parallel p late capacitor, in air in such a way that one of their edges is perpendicular, to an oil surface in a tank fil led with an insulating oil. The plates are connected to a battery of e.m.f. 500 volt. The plates are then lower ed vertically into the oil at a speed of 0.001 m/s. Calculate the current drawn from the battery during the pr ocess, [di-electric constant of oil = 11, e 0 = 8.85 x 10" 1 2 C 2/ N 2 m 2 ] [ JEE '94, 6 ] A parallel plate capacitor C is connected to a battery & is charged to a potenti al difference V. Another capacitor of capacitance 2C is similarly charged to a potential difference 2V volt. The charging batteiy is now disconnected & the capacitors are connected in parallel to each other in suc h a way that the positive terminal of one is connected to the negative terminal of other. The final ener gy of the configuration is: (A) zero (B) - CV 2 25 (C) CV 2 (D) - CV 2 [JEE' 95, 1 ] The capacitance of a parallel plate capacitor with plate area 'A' & separation d is C. The space between the plates is filled with two wedges of di-electric co nstant Kj & K2 respectively. Find the capacitance of the resulting capacitor. [JEE' 96, 2] IOOV 2nF 1 - . Two capacitors A and B with capacities 3 pF and 2 pF are charged to a potential difference of 100 V and 180 V respectively. The plates of the capacitors are connected as shown in figure with one wire from each capacitor free. The upper plate of a is positive and that of B is negative, an uncharged 2 pF capacitor C with lead wires falls on the free ends to complete the circuit. Calculate: the final charges on the three capacitors The amount of electrostatic energy stored in the system before and after the com pletion of the circuit. [JEE' 97 (cancelled)] B 180V /.o/ c/. c+t+ct:+ct ::: Q.6 An electron enters the region between the plates of a parallel plate capaci tor at a point equidistant from eitherplate. The capacitor plates are 2* 10 _ 2 mapart & 10 - 1 m long. A potential difference of300 volt is kept across the plates. Assuming that the initial velocity of the electron is parallel to the capacitor plates, calculate the largest value ofthe velocity of the electron so that they do not fly out of the capacitor at the other end. [ JEE '97, 5 ] Q. 7 For the circuit shown, which of the following statements is true ? (A) with S, closed, Vj = 15 V, V2 = 20 V (B) with S3 closed, Vj = V2 = 25 V (C) with & S2 closed, Vj = V2 = 0(D) with Sj & S2 closed, Vl = 30 V, V2 = 20 V V, =30V [JEE' 99, 2] Q.8 Calculate the capacitance of a parallel plate condenser, with plate area A and distance between plates d, when filled with a medium whose permittivity varies as; e (x)= e 0 + P x S ( X ) = G 0 + P ( d - x ) 0 < x < | 4 < x < d. [REE2000, 6] Q. 9 Two identical capacitors, have the same capacitance C. One of them is char ged to potential Vt and the other to V2 . The negative ends of the capacitors are connected together. When t he positive ends are also connected, the decrease in energy of the combined system is [ JEE 2002 (Scr), 3 ] (A) Mvf-vl) (B)Mv, 2 +v 2 2 ) ( q I c ^ - v J ( D ^ c f a +v J Q.10 In the given circuit, the switch S is closed at time t = 0. The charge Q o n the capacitor at any instant t is given by Q (t) = Q0 (l-e" 0 *). Find the value of Q0 and a in terms of given parameters shown in the circuit. [JEE 2005] Q. l l Given: Rj = ID , R 2 = 2Q, C x = 2pF, C2 = 4pF The time constants (in pS) for the circuits I, n, HI are respectively .C, !!i hi "C 2 k v h- R,: K r . - T T-r. , v V ' (II.) (A) 18, 8/9, 4 (C) 4, 8/9, 18 (in) (B) 18, 4, 8/9 (D) 8/9, 18,4 1m s/ + c^ "T v [JEE 2006]/.o/ c/. ::: ANSWER KEY EXERCISE # Q.2 (a) 20 Q.4 30 V 32 Q.8 - M F Q.5 C 1 q 2 d Q 6 i Z T Q.l 9J Q.3 3Q/2C Q.7 0 100 Q.9 (a) c+t+ct:+ct I pC, (b) 0.3 mJ, (c) 0.6 mJ. (d) 60 [iCvolts; (b) 28.56 |iC, 42.84 pC, 71.4 jnC, 22.88 pC Q.10 10 pCQ. l l 25 eA 24 d Q.15 150 mJ Q.12 ^ F Q.13 A G0 V 2A e 0 V Q.14 60 (i c, At oB Q.16 (i) | CV 2 ; (ii) - ~ CV2(K- 1); ^ (K + 2) (K - l ^ V 2 ; q(nC), 200 Q. 17 (i) 5 X 10 4 V/m, 3 x 10 4 V/m; (ii) 3 5/9 Q.18 EXERCISE # II HHI Q.l (a) 12 V, 9 V, 3 V, 13 V, 16 V, (b) m T TT Q.2 C = GSTIK, 2d Q.3 (i) 3 5 f e 0 A^ ;(ii)Q 3 =T I AV ,Q 5 = t v " y ,AVQ.4 (i) 1.5 x 10 4 V/m, 4.5 x 10 4 V/m, (ii) 75 V, 225 V, 7 C/m 2 Q.5 2[iA Q.6 W = \ 2 1K Q.7 C = Cn q.8 qi - Ci 2 V(C 2 +C 3 ) Ci C2 +C2 C3 +Cj c 3 C1 C2 C3 V / : : v o V V -1 = 0.01078 |iF,n = c1 c2 +c2 c3 +c3 c1 SpAV 2 2d 2 (d0 -d!)'v As0 ^3 \ 3/ 2 Kdf-e0 AV 2 MDJ 1/2 Q. l l 12 volt Q.13 69 Q.14 U, 3kg, 2 lOr where q, = ~ ; U u = 2K(q +q i ) 2 / 3 5 r Q.15 q = CE r O CV 1 v e y + Q.16 q CV 1-e~(iii) 8 x 10"C0 V020mCt/RC 2 /.o/ c/. c+t+ct:+ct Q.17 For t < 250 ps, 0 0 t amp ; For t > 250 ps, I = 6 a m p ; I(ajnp) 0.04 0.015 4 0 0 ^ Q . 1 8 - P -o.n t ( x I O^ s ) EXERCISE # III Q.l (i) 0.2 x 10" 8 F, 1.2 x lO" 5 J ; (ii) 4.84 x 10" 5 J ; (iii) 1.1 x 10" 5 J Q.2 4.425 x 10~ 9 Ampere Q . 3 B q.4 CK^ /n K, (Ka-KO K, Q. 5 Q A = 9 0 pC, Q B = 1 5 0 pC, Q C = 2 1 0 pC, UJ = 4 F = 1 8 MJ V48 Q ' 6 2^9A & Q.9 C Q.7 D Q.8 ^ 2 e 0 2 s 0 CVR, R1+R2 Q. 10 Q 0 = R::: I = 0.04 e^ 0.1 ie-4000(t-250)xi(r7 . 4 MJ, Ui + R 2 anda= Q. l l XII (ALL) quesjjommm. R2 i f E, = E2 ( B) C1 < C2 i f E 1 = E 2 (C) RjCJ > R. C, (D) f 1 < ^ 2 C 1 Q.13 Aparallel plate capacitor is charged by connecting it to a battery. The ba ttery is disconnected and the plates of the capacitor are pulled apart to make the separation between the plat es twice. Again the capacitor is connected to the battery (with same polarity) then (A) Charge from the battery flows into the capacitor after reconnection (B) Charge from capacitor flows into the battery after reconnection. (C) The potential difference between the plates increases when the plates are pu lled apart. (D) After reconnection of battery potential difference between the plate will im mediately becomes half of the initial potential difference. (Just after disconnecting the battery) Q. 14 The plates of a parallel plate capacitor with no dielectric are connected to a voltage source. Now a dielectric of dielectric constant K is inserted to fill the whole space between the plates with voltagesource remaining connected to the capacitor. (A) the energy stored in the capacitor will become K-times (B) the electric field inside the capacitor will decrease to K-times (C) the force of attraction between the plates will increase to K 2 -times (D) the charge on the capacitor will increase to K-times Q. 15 Four capacitors and a batteiy are connected as shown. The potential drop across the 7 pF capacitor is 6 V. Then the : J H (A) potential difference across the 3 pF capacitor is 10 V (B) charge on the 3 pF capacitor is 42 pC (C) e.m.f. of the battery is 3 0 V (D) potential difference across the 12 pF capacitor is 10 V. 3.9(.IF J7nF "puF Q. 16 A circuit shown in the figure consists of a battery of emf 10 V an d two capacitance C, and C2 of capacitances 1.0 pF and 2.0 pF respectively. The potential difference V A - VB is 5 V A o | | | | | | o B (A) charge on capacitor Cj is equal to charge on capacitor C2 (B) Voltage across capacitor Cj is 5V. c' e q, (C) Voltage across capacitor C2 is 10 V (D) Energy stored in capacitor C. is two times the energy stored in capacitor C2 . Q.17 A capacitor C is charged to a potential difference V and batteiy is discon nected. Now if the capacitor plates are brought close slowly by some di stance: (A) some +ve work is done by external agent (B) energy of capacitor will decrea se (C) energy of capacitor will increase (D) none of the above /. o/ c/. o.. o/ c. ::: Q.18 The capacitance of a parallel plate capacitor is C when the region between the plate has air. This region is now filled with a dielectric slab of dielectric constant k. The capacitor is connected to a cell of emf E, and the slab is taken out (A) charge CE(k - 1 ) flows through the cell (B) energy E 2 C(k - 1) is absorbed by the cell. (C) the energy stored in the capacitor is reduced by E 2 C(k - 1 ) (D) the external agent has to do ^E 2 C( k - 1 ) amount ofwork to take the slab out. Q.19 Two capacitors of capacitances 1 pF and 3 pF are charged to the same volta ges 5 V. They are connected in parallel with oppositely charged plates connected together. Then: (A) Final common voltage will be 5 V (B) Final common voltage will be 2.5 V (C) Heat produced in the circuit will be zero. (D) Heat produced in the circuit will be 37.5 pJ Q. 20 The two plates X and Y of a parallel plate capacitor of capacitance C are given a charge of amount Q each. X is now joined to the positive terminal and Yto the negative terminal of a cell of emf E = Q/C. (A) Charge of amount Q will flow from the negative terminal to the positive term inal of the cell inside it (B) The total charge on the plate X will be 2Q.(C) The total charge on the plate Y will be zero. (D) The cell will supply CE 2 amount of energy. Q.21 A dielectric slab is inserted between the plates of an isolated charged ca pacitor. Which of the following quantities will remain the same? (A) the electric field in the capacitor (B) the charge on the capacitor (C) the potential difference between the plates (D) the stored energy in the ca pacitor. Q.22 The separation between the plates of a isolated charged parallel plate cap acitor is increased. Which of the following quantities will change? (A) charge on the capacitor (B) potential difference across the capacitor (C) energy of the capacitor (D) energy density between the plates. Q.23 Each plate of a parallel plate capacitor has a charge q on it. The capacit or is now connected to a battery. Now, (A) the facing surfaces of the capacitor have equal and opposite charges. (B) the two plates of the capacitor have equal and opposite charges. (C) the battery supplies equal and opposite charges to the two plates. (D) the outer surfaces of the plates have equal charges. Q. 24 Following operations can be performed on a capacitor: X - connect the capacitor to a battery of emf E. Y - disconnect the battery Z - reconnect the battery with polarity reversed. W - insert a dielectric sla b in the capacitor (A) In XYZ (perform X, then Y, then Z) the stored electric energy remains unchan ged and no thermal energy is developed. (B) The charge appearing on the capacitor is greater after the action XWY than after the action XYW. (C) The electric energy stored in the capacitor is greater after the action WXY than after the action XYW. (D) The electric field in the capacitor after the action XW is the same as that after WX. Q.25 A parallel plate capacitor is charged and then disconnected from the sourc e of potential difference. If the plates of the condenser are then moved farther apart by the use of insulated han dle, which one of the following is true? (A) the charge on the capacitor increases (B) the charge on the capacitor decre ases (C) the capacitance of the capacitor increases (D) the potential difference acr oss the plate increases /. o/ c/. o.. o/ c. ::: Q.26 Aparallel plate capacitor is charged and then disconnected from the source steady E.M.F. The plates are then drawn apart farther. Again it is connected to the same source. Then: (A) the potential difference across the plate increases, while the plates are be ing drawn apart. (B) the charge from the capacitor flows into the source, when the capacitor is r econnected. (C) more charge is drawn to the capacitor from the source, during the reconnecti on. (D) the electric intensity between the plates remains constant during the drawin g apart of plates. Q.27 When a parallel plates capacitor is connected to a source of constant pote ntial difference, (A) all the charge drawn from the source is stored in the capacitor. (B) all the energy drawn from the source is stored in the capacitor.(C) the potential difference across the capacitor grows very rapidly initially a nd this rate decreases to zero eventually. (D) the capacity of the capacitor increases with the increase of the charge in t he capacitor. Q.28 When two identical capacitors are charged individually to different potent ials and connected parallel to each other, after disconnecting them from the source: (A) net charge on connected plates is less than the sum of initial individual ch arges. (B) net charge on connected plates equals the sum of initial charges. (C) the net potential difference across them is different from the sum of the in dividual initial potential differences. (D) the net energy stored in the two capacitors is less than the sum of the init ial individual energies. Q. 29 Aparallel plate capacitor of plate area A and plate seperation d is charg ed to potential difference V and then the battery is disconnected. A slab of dielectric constant K is then insert ed between the plates of the capacitor so as to fill the space between the plates. If Q, E and W denote respe ctively, the magnitude of charge on each plate, the electric field between the plates (after the slab is i nserted) and the work done on the system, in question, in the process of inserting the slab, then e0 AV s0 KAV V AV 2 1 - 1 K Q. 3 0 A parallel plate capacitor is connected to a battery. The quantities cha rge, voltage, electric field and energy associated with the capacitor are given by Q0 , VQ, E0 and U0 respectiv ely. A dielectric slab is introduced between plates of capacitor but battery is still in connection. The c orresponding quantities now given by Q, V, E and U related to previous ones are ( A) Q>Q0 (B) V > V0 (C) E > Eq ( D) U< U0 Q.31 A parallel-plate capacitor is connected to a cell. Its positive plate A an d its negative plate B have charges +Q and - Q respectively. A third plate C, identical to A and B, with charge +Q, is now introduced midway between A and B, parallel to them. Which of the following are correct? 3Q (A) The charge on the inner face of B is now (B) There is no change in the potential difference between A and B. (C) The potential difference between A and C is one-third of the potential diffe rence betweenB and C. (D) The charge on the inner face of A is now Q/ 2. Q.32 Two capacitors Cj = 4 pF and C2 = 2pF are charged to same po tential V = 500 Volt, but with opposite polarity as shown in the figure. The switches St and S2 are closed. (A) The potential difference across the two capacitors are same and is given by 500/3 V (B) The potential difference across the two capacitors are same and is given by 1000/3 V (C) The ratio of final energy to initial energy of the system is 1/9. (D) The ratio of final energy to initial energy of the system is 4/9. /. o/ c/. o.. o/ c. ::: Q. 33 A parallel plate capacitor is charged to a certain potential and the charging battery is then disconnected. Now, if the plates of the capacitor are moved apart then: (A) The stored energy of the capacitor increases (B) Charge on the capacitor increases (C) Voltage of the capacitor decreases (D) The capacitance increases Q. 34 If a battery of voltage V is connected across terminals I of the block b ox shown in figure, an ideal voltmeter connected to terminals II gives a reading of V/2, while if the battery is connected to terminals II, a voltmeter across terrninals I reads V. The black box may contain (A) i O-J I ! OIvwvl R i R ' R OIvwv(C) ER Oivwvl R -o 11 - o 11 -o (B) 1 1 c 1 T (D) 1 . i i 1 i 1 1 j) J ! T 1 Q.35 Two capacitors of equal capacitance (Cj = C2 ) are shown in the figure. Initially, while the switch S is open, one of the capacitors is uncharged and the other carries charge Q0 . The energy stored in the charged capacitor is U0 . Sometimes after the switch is closed, the capacitors Cj and C2 carry charges Qj and Q2 , respectively; the voltages across the capacitors are V{ and V2 ; and the energies stored in the capacitors are Uj and U2 . Which of the following statements is INCORRECT ? 4= Co (A) Q 0 = - (Qj + Q 2 ) ( C) Vj =V2( E) U0 = Uj + U2 (B) Qj = Q 2 ( D) Uj = U2 Question No. 3 6 to 39 (4 questions) The figure shows a diagonal symmetric arrangement of capacitors and a battery Q. 3 6 Identify the correct statements. (A) Both the 4pF capacitors carry equal charges in opposite sense. (B) Both the 4pF capacitors carry equal charges in same sense. ( C ) V B - V D > 0 ( D ) V d - V B > 0 fi 2\xF h T 2(xF 2 (iF 4|iF E=20V Q. 3 7 If the potential of C is zero, then (A) VA = + 20 V ( C) 2 ( VA - Vd ) + 2 ( VB - Vd ) = 4VD ( B) 4 ( VA - VB ) + 2 ( VD - VB ) = 2VB ( D) VA = VB + VD Q. 3 8 The potential of the point B and D are (A) VB = 8 V (B) VB = 12V (C) VD = 8 V ( D) Vd =1 2 V /. o/ c/. o.. o/ c. ::: Q.39 The value of charge q1 ; q2 and q3 as shown in the figure are (A) qj = 32 pC ; q2 = 24 pC ; q3 = - 8 pC (B) q{ = 48 pC ; q2 = 16 pC ; q3 = + 8 pC (C) qj = 32 pC ; q2 = 24 pC ; q3 = + 8 pC (D) q( = 3 pC ; q2 = 4 pC ; q3 = + 2 pC qi -HP B 12 i-l ^ q2 D E=20V qi Q.40 If Q is the charge on the plates of a capacitor of capacitance C, V the po tential difference between the plates, A the area of each plate and d the distance between the plates, the force of att raction between the plates is (A) v 7 2 V 8 o A (B) r CV 2 A ) 'a(q)'a00 rb o rb 66 8I"b 7 v srb SITTINGII Tl oi x n orb sASgi-oixzrt? z,rb r> V H'6 ra t>-0l x t? l'b 001 '0i7 '0 '0 '09 'SL '00 '081' % SZ 0lb < u 'Z/(.~ u ) (fr-n) 8'b o Kb a z/b V 'b V 9"b o rb 9'862 6l'b V 9rb S/Da01x8> irb X Jxyu^v T-x D y(q) V00 6 b sb rb 'Ii o/ c/. t+// o./ u/.. +.-. ::cttctt+t o:to rotr tortt tttc: txttctstt Q. 1 The bob of a simple pendulum of length I is released from point P. What is the angle made by the net acceleration of the bob with the string at point Q. Q.2 Aballofmass 1 kg is released from position A inside a wedge with a hemisphe rical cut of radius 0.5 m as shown in the figure. Find the force exerted by the vertic al wall OM on wedge, when the ball is in position B. (neglect friction everywhere). Take(g = 10m/s 2 ) Q.3 A particle P is moving on a circle under the action of only one force actin g always towards fixed point O on the circumference. Find ratio of / / / / / / / / / / / / / / / / ere dt 2 & 'de^ 2 v dt j Q.4 A particle is moving in x direction, under the influence of force F = 7T si n nx. Find the work done by another external agent in slowly moving a particle from x = 0 to x = 0.5 m. Q.5 A particle moves in a circle of radius R with a constant speed v. Then, fin d the magnitude of average 71R acceleration during a time interval ? y . u m u u u I k Q.6 In the figure shown, pulley and spring are ideal. Find the potential energy stored in the spring (m, > m2 ). Q.7 A spring of mass m is pulled such that a given instant,, velocity of both o f its end is v in the opposite direction. Find the kinetic energy of the spring. Q.8 A particle of mass 3 kg is rotating in a circle of radius 1 m such that the angle rotated by its radius is given by 0 = 3 (t + sint). Find the net force acting on the particle when t = n/2. Q.9 For a particle rotating in a vertical circle with uniform speed, the maximu m and minimum tension in the string are in the ratio 5 :3. If the radius of vertical circle is 2m, then find the speed of revolving body. Q.10 Two strings of length /=0.5 m each are connected to a block of mass m=2 k g at one end and their ends are attached to the point A and B 0.5 m apart on a vertic al pole which rotates with a constant angular velocity co=7 rad/sec. Find the ratio T, 0.5 of tension in the upper string (T,) and the lower string (T2). [Use g = 9.8 m/ s 2 ] Q. l l A force F = - k( x i + y j) [where k is a positive constant] acts on a particle moving in the x-y plane. Starting from origin, the particle is taken to (a, a) and then to (a/V2,o). Find the total work done by the force F on the particle. , o/ c/. t/. t.- ::: Q.12 A bead of mass m is attached to one end of a spring of natural length -J3 R and (V3 +l ) mg spring constant k= R . The other end of the spring is fixed at point A (60 Q.13 Q.14 Q.15 on a smooth fixed vertical ring of radius R as shown in the figure. What i s the normal reaction at B just after the bead is released? Water is pumped from a depth of 10 m and delivered through a pipe of cross sect ion 10~ 2 m 2 upto a height of 10 m. If it is needed to deliver a volume 0.2 m 3 per second, find the power required. [Use g= 10m/ s 2 ] A mass m rotating freely in a horizontal circle of radius 1 m on a fiictionless smooth table supports a stationary mass 2m, attached to the other end of the string passing through smooth hole O in table, hanging t vertically. Find the angular velocity of rotation. 2m Consider the shown arrangement when a is bob of mass' m' is suspended by means of a string connected to peg P. If the bob is given a horizontal velocity u ha ving magnitude N/3g7, find the minimum speed of the bob in subsequent motion. Q.16 A bead of mass m is tied at one end of a spring of spring constant R mg R and unstretched length and other end to fixed point O. The smooth semicircular wire frame is fixed in vertical plane. Find the normal reaction between bead and wire just before it reaches the lowest point. Q.17 A particle of mass m is hanging with the help of an elastic string of unst retched length a and force mg constant . The other end is fixed to a peg on vertical wall. String is given an additional extension of a 2a in vertical downward direction by pulling the mass and released from rest. Find the maximum height reached by it during its subsequent motion above point of release. (Neglect inte raction with peg if any) Q.18 A particle of mass 1 kg is given a horizontal velocity of 4 m/s along a h orizontal surface, with which it has a coefficient of friction (both static and kinetic) of 0.4. The particle strikes a fixed ideal spring of force constant 6 N/m after travelling a distance of 0.25 m. Assume acceleration due to gravity is 10 m/s 2 . Find the final displacement of the particle from its starting point. Q.19 A particle P of mass m is placed inside a hemispherical bowl which rotates about its vertical axis with constant angular velocity co. The particle is just prevented from sliding down when the radius vector OP joining it to the centre of the bowl O makes an angle of 45 with the axis. The ra dius of the bowl is \ 0V2 and the coefficient of friction between the particle and the bowl is 0.5. Find the value of angular velocity co. Q.20 A point moves along a circle having a radius 20 cm with a con stant tangential acceleration 5 cm/s 2 . How much time is needed after motion begins for the normal acceleration of the point to be equal to tangential acceleration ? e=10m/s ! 4 m/s | 1 kg | |i=0.4 0.25m s,o/ c/. t/. t.- ::: Q.21 A body of mass 2 kg is moving under the influence of a central force whose potential energy is given by U (r) = 2r 3 Joule. If the body is moving in a circular orbit of 5m,then find its energy. y Q.22 A ring rotates about z axis as shown in figure. The plane of rotation i s xy. At a certain instant the acceleration ofa particle P (shown in figure) on the ring is (6 i-8 j) m/s 2 . find the angular acceleration of the ring & the angular velocity at that instant. Radius of the ring is 2m. Q.23 A particle is revolving in a circle of radius lm with an angular speed of 12 rad/s. At t = 0, it was subjected to a constant angular acceleration a and its angular speed increased to(480/7i) rpm in 2 sec. Particle then continues to move with attained speed. Calculate (a) angular acceleration of the particle, (b) tangential velocity of the particle as a function of time. (c) acceleration of the particle at t = 0.5 second and at t = 3 second (d) angular displacement at t = 3 second. Q.24 The member OA rotates in vertical plane about a horizontal axis through O with a constant counter clockwise velocity co = 3 rad/sec. As it passes the position 9 = 0, a small mass m is placed upon it at a radial distance r=0.5 m. If the mass is ob served to slip at 0=37, find the coefficient of friction between the mass & the member. Q.25 AparticlePis sliding down a fiictionless hemispherical bowl. It passes the point A at t =0. At this instant of time, the horizontal component of its velocity is v. A bead Q of the same mass as P is ejected from A at t=0 along the horizontal string AB, with the speed v. Friction between the bead and the string may be neglected. Which bead reaches point B earlier? txtt ctsttt Q. 1 A particle is confined to move along the +x axis under the action ofa forc e F(x) u that is derivable from the potential U(x) =ax 3 -bx. (a) Find the expression for F(x) (b) When the total energy of the particle is zero, the particle can be trapped with in / the inteivalx= o to x= x.. For this case find the values of x,. (c) Determine the maximum kinetic energy that the trapped particle has in its m otion. Express all answers in terms a and b. Q.2 A particle of mass 2kg is subjected to a two dimensional conserv ative force given by Fx =-2x+2y, Fy=2x-y 2 . (x,y in m and F in N) If the particle has kinetic energy of (8/3) J at point ( 2,3), find the speed of the particle when it reaches (1,2). Q.3 A square plate is firmly atached to a fiictionless horizontal plane. One en d of a taut cord is attached to point A of the plate and the other end is attached to a sphere of mass m. In the process, the cord gets wrapped around the plate. The sphere is given an init ial velocity vQ on the horizontal plane perpendicular to the cord which causes it to make a complete circuit of the plate and return to point A. Find the velocity of the sp here when it hits point A again after moving in a circuit on the horizontal plane. Also fi nd the time taken by the sphere to complete the circuit. Q.4 A coin is placed on the horizontal surface of a rotating disc, If the disc starts from rest and is given a constant acceleration a = l/V2 rad/s 2 , find the number ofrevolution through which the disc turns before the coin slips. The distance of coin from axis is 1 m initially and the coeffici ent of friction p=0.5. , o/ c/. t/. t.- ::: Q.5 (i) Q.6 Q.7 (a) (b) (c) Q.8 Q.9 Q. l l (i) O Q. 10 A small bead of mass m is free to slide on a fixed smooth vertical wire, as indicated in the diagram. One end of a light elastic string, of unstretched length a and f orce constant 2mg/a is attached to B. The string passes through a smooth fixed ring R and the other end of the string is attached to the fixed point A, AR being horizonta l. The point O on the wire is at same horizontal level as R, and AR=RO = a. In the equilibrium position, find OB. The bead B is raised to a point C of the wire above O, where OC = a, and is rele ased from rest. Find the speed of the bead as it passes O, and find the greatest depth below O of the bea d in the subsequent motion. A particle of mass 5 kg is free to slide on a smooth ring of radius r = 20 cm f ixed in a vertical plane. The particle is attached to one end of a spring whose other e nd is fixed to the top point O of the ring. Initially the particle is at rest at a po int A of the ring such that Z OCA=60, C being the centre of the ring. The natural length of the spring is also equal to r = 20cm. After the particle is released and slides dow n the ring the contact force between the particle & the ring becomes zero when it reac hes the lowest position B. Determine the force constant of the spring. A small block of mass m is projected horizontally from the top of the smoo th hemisphere of radius r with speed u as shown. For values of u > uQ, it does not slide on the hemisphere (i.e. leaves the surface at the top itself). For u = 2u0, it lands at point P on ground Find OP. For u = u0/3, Find the height from the ground at which it leaves the hemisphere. Find its net acceleration at the instant it leaves the hemisphere. The track in Fig is straight in the horizontal section AB and is a semicircle of radius R in the vertical part BCD. A particle of mass m is given a velocity of A/(22gR)/5 to the left along the track. The particle moves up the vertical sect ion and ultimately loses contact with it. How far from point B will the mass land. A small particle of mass 1 kg slides without friction from height H=45 cm shown in figure and then loops the vertical loop of radius R from where a section of angle 6 = 60 has been removed. Find R such that after losing contact at A and flying through the air, the particle will reach at the point B. Also find the normal reaction between particle and path at A. A ring of mass m slides on a smooth vertical rod. A light string is attached to the ring and is passing over a smooth peg distant a from the rod, and at the other end of the string is a mass M (> m). The ring is held on a level with the peg and released: Show that it first comes to rest after falling a distance: 2mMa 2 2 M - m JZL M=0 l i l t 7 M777 Ablock ofmass m is held at rest ona smooth horizontal floor. Alight fiictionless , small pulley is fixed at aheight of 6 m from the floor. Alight inextensible stri ng of length 16 m, connected with Apasses over the pulley and another identical block B is hung from the string. Initial height of B is 5m from the floor as shown in Fig. When the system is released from rest, B starts to move vertically downwards and A slides on the floor towards right. If at an instant string makes an angle 0 with horizontal, calculate relation bet ween velocity u of A and v of B Calculate v when B strikes the floor. 6 m , o/ c/. t/. t.- ::: Q.13 (a) (b) Q.14 Q.15 Q.16 v=! const./ Q.12 A small block can move in a straight horizontal linea along AB. Flash ligh ts from ^ one side projects its shadow on a vertical wall which has horizontal cross secti on as a circle. Find tangential & normal acceleration of shadow of the block on the ^ wall as a function of time if the velocity of the block is constant (v). 0, the functional form of the pot ential energy U (x) of the particle is [JEE (Scr.)'2002] U(x) (A) U(x) (B) U(x)f X (C) U(x) X (D) X Q.13 An ideal spring with spring-constant k is hung from the ceiling and a bloc k of mass M is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is [JEE (Scr.)'2002] (A) 4 Mg/k (B) 2 Mg/k (C)Mg/k (D)Mg/2k Q.14 A spherical ball of mass m is kept at the highest point in the space betwe en two fixed, concentric spheres Aand B (see figure). The smaller sphere A has a radius R and the space between the two spheres has a width d. The ball has a diameter vei y slightly less than d. All surfaces are frictionless. The ball is given a gentle push (towards the right in the figure). The angle made by the radius vector of the ba ll with the upward vertical is denoted by 9 (shown in the figure). [JEE' 2002] (a) Express the total normal reaction force exerted by the spheres on the ball as a function of angle 9. (b) Let Na and NB denote the magnitudes of the normal reaction force on the b all exerted by the spheres A and B, respectively. Sketch the variations of NA and NB as functions of cos0 i n the range 0 < 9 < TT by drawing two separate graphs in your answer book, taking cos9 on the horizontal a xes. Sphere B Sphere A Q.15 In a region of only gravitational field of mass 'M' a particle is shifted from Ato B via three different paths in the figure. The work done in different paths are W,, W2 , W3 respectively then [JEE (Scr.)'2003] (A) W, = W2 = W3 (C) Wj > W~ > w3 (B) W, (D) Wi < W2 < W3 = w 2 > w3 Q.16 A particle of mass m, moving in a circular path of radius R with a constant speed v2 is located at point (2R, 0) at time t = 0 and a man sta rts moving with a velocity v, along the +ve y-axis from origin at time t =0. Calculate the linear momentum of the particle w.r.t. the man as a function oftime. [JEE 2003] (0,0) VL V2 Q.17 A particle is placed at the origin and a force F = kx is acting on it (whe re k is a positive constant). If U(0)=0, the graph of U(x) versus x will be (where U is the potential energy func tion) U(x) (A) U(x) (B) U(x) (C) U(x) (D) [JEE' 2004(Scr)] , o/ c/. t/. t.- ::: CENTRE OF MASS MOMENTUM & COLLISION o ~t Graphically, impulse is the area under the F-t graph The action of force with respect to time is defined in terms of Impulse, that is , 1= j*Fdt = mvf - mvi =Ap In the absence of a net external force, the momentum of a system is conserved. dP ^ =F e*t = 0 p = Pj + p2 + + p N = constant 1. Collision is a kind of interaction between two or more bodies which come in contact with each other for a very short time interval. 2. Types of collision: Elastic and Inelastic Collisions may be either elastic or inelastic. Linear momentum is conserved in b oth cases. (i) A perfectly elastic collision is defined as one in which the total kinetic energy of the system is conserved. (ii) In an inelastic collision, the total kinetic energy of the system changes. (iii) In a completely inelastic collision, the two bodies couple or stick togeh ter. 3. Coefficient of Restitution : It is defined as the ratio of the velocity of s eparation to the velocity of approach of the two colliding bodies. rel. velocity of separation rel. velocity of approach For a perfectly elastic collision, e = 1 For an inelastic collision, 0 < e < 1 For completely inelastic collision, e = 0 Note that the velocity of approach and the velocity of separation are always tak en along the normal tothe striking surface. CENTRE OF MASS y f r. 1. Discrete System : The position vector of the centre of mass is m1 r , +m2 r 2 + + mn r n y ' H \ 3 r c m! + m2 + mn ni4 T HI3 where fj, r2 ,..., rn are the position vectors of masses mp m2 ,..... mn resp ectively. The components of the position vector of centre of mass are defined as Z m i x i . _ Z m .yi . _ Z m i z i M ' Y c M ' Z M X c = 2. Continuous system: The centre of mass of a continuous body is defined as rc = fr dm c M J In the component form fx dm v = fydm z = fzdm M J ' c M J ' c M J M , o/ c/. t/. t.- ::: 3. Centre of Mass of Some Common Systems : (i) A system of two point masses. The centre of mass lie closer to the heavier mass. (iv) 4. (iii) (iv) 5. I* im2 M m, +m ni i +mj(ii) A circular cone h 4 yc (iii) A semi-circular ring 2R y c = ; x = 0 c TI c A semi-circular disc 4R 0! X* yc = 3tt ; x =o (v) A hemispherical shell R y c = 7 ; x c =0 (vi) A solid hemisphere 3R 0 ' Motion of the centre of mass : Velocity: The instantaneous velocity of the centre of mass is defined as X m i v i v c M Acceleration: The acceleration of the centre of mass is defined as X m i a > a c = M Momentum : The total momentum of a system of particles is p = Mvc Kinetic Energy: The kinetic energy of a system of particles consisits of two par ts. K = Kc + K' 1 2 where Kc - Mvc , kinetic energy due to motion of c.m. relative to the fixed or igin O, V- 1 2 and K' = 2_, ^ m iv i > kinetic energy of the particles relative to the c.m. Note that the term K' may involve translational, rotational or vibrational energ ies relative to the centre of mass. Newon's Laws of a system of particles: The first and second laws of motion for a system of particles are modified as: First law: The centre of mass of an isolated system is at rest or moves with con stant velocity. Second law: The net external force acting on a system of total of mass M is rel ated to the acceleration of centre of mass of the system. I S ext M< l c m , o/ c/. t/. t.- ::: ct:tt ot +ss ot:t cotttsto txttctstt Q.l A hemisphere of radius R and of mass 4m is free to slide with its base on a smooth horizontal table. A particle of mass m is placed on the top of the hemisphere. F ind the angular velocity of the particle relative to hemisphere at an angular displa cement 0 when velocity of hemisphere has become v. Q.2 A man whose mass is m kg jumps vertically into air from a sitting position in which his centre of mass is at a height hj from the ground. When his feet are just about to leave the ground his centre of mass is h2 from the ground and finally rises to h3 when he is at the top of the jump, (a) What is the average upward force exerted by the ground on him? (b) Find work done by normal reaction from g round. Q.3 In the figure shown, each tiny ball has mass m, and the string has length L . One of the ball is imparted a velocity u, in the position shown, in which the initial d istance Q. 4 Two trolleys A and B are free to move on a level fiictionless track, and a re initially stationary. A man on trolley A throws a bag of mass 10 kg with a horizontal velocity of 4 m/s with re spect to himself on to trolley B of mass 100 kg. The combined mass of trolley A (excluding bag) and the man is 140 kg. Find the ratio of velocities of trolleys A and B, just after the bag lands on trolley B. Q.5 A bob of mass m attached with a string of length I tied to a point on ceili ng is released from a position when its string is horizontal. At the bottom most point of its motion, an identi cal mass m gently stuck to it. Find the angle from the vertical to which it rises. Q.6 Tw