Irfan Ali KhanLecturerGC CCW0345-9998898

CHAP 2

VECTORS AND EQUILIBIRIUM

Q.1 Define the following terms:(i) Unit vector (ii) Position vector (iii) Component of vector

Ans: (i) Unit vector: A vector whose magnitude is one is called unit vector. It is usually used to represent the direction of a vector. A unit vector can be calculated by dividing a vector by its

magnitude.

(ii) Position vector: A vector which describes the postion of a point relative to origin. The position vector of a point P(a,b) in X-Y plane is given by

The magnitude of r is given by

Similarly the position vector r of a point P(a,b,c) in space is given by

and magnitude of r is given by

(iii) Component of vector: A component of a vector is its effective value in a given direection. A vector is comsidered as the resultant of its components vectors along the specified directions.

Q.2 The vector sum of three vectorrs gives a zero resultant. What can be the orientation of the vectors.Ans. If three victors are such that they can be represented by the three sides of a triangle taken in cyclic order, then the vector sum of the three vectors will be zero as shown in fig.

Q.3 vector lies in XY Plane. For what orientation

will both of its rectangular components be negative? For what orientation will its components have opposite signs?Ans, when vector lies in thrid quadrant its both components will negative. When vector lies in second or forth quadrant its components will have opposite signsQ.4 If one component of a vector is not zero, can its magnitude be zero?Ans. No its magnitude can never be zero. As the magnitude of a vector A is

This equation shows that magnitude of A will be zero if all of its components are zero.Q.5 Can a vector have component greater than vector’s magnitude?Ans. No component of a vector can never be greater than its magnitude.

If and are rectangular of components of A then

Short Questions

Thus a vector’s component may be equal in some cases to vector’s magnitude but can never be greater than itQ.6 Can the magnitude of vector have a negative valueAns. No magnitude of a vector can not be negative. According to equation magnitude of vector is

As squar of a real number is always positive so magnitude can not be negative. Moreover sometimes negative sign with magnitude reffered to the direction of vector.

Q.7 If What can you say about the

components of the two vectors.

Ans. As or

So and

Thus rectangular components of the given vectors are equal in magnitude and directed opposite to each other as shown.Q.8 Under what circumstances would a vector have components that are equal in magnitude.Ans. Components of vector will be equal if vector is at 450 1350 or 2250

with X- axis. When rectangular components of vector are

It shows that components will be equal if vector maken an angle of 450 with X-axis Q.9 Is it possible to add a scaler quantity to vector quantity. Explain?Ans. No its not possible to add a scaler quantity to a vector quantity.Because vectors are added using head to tail rule and scalers are added using simple arithmatic rules. As they have different natures so scalers can not be added to vectorsQ.10 Can you add zero to a null vector?Ans. No its not possible to add a zero to a null vector.As zero is a scaler quantity and null vector is vector quantity and acalers can not be added bo vectors so its not possible to add zero to null vector.Q.11 Two vecotrs have unequal magnitude. Can there sum be zero? Explain?Ans. No their sum can not be zero.Sum of two vectors can only be zero if they have same magnitude and opposite direction. Two vectors of unequal magnitude can not be combined to give a zero resultant.Q.12 Show that sum and difference of two perpendicular vectors of same magnitude are perpendicular and of same length. Ans. Consider two vectors and of same magnitude and are

perpendicular to each other as shown. Thus

We are to prove that is perpendicular to so we will

prove that

Now

So sum and difference of two perpendicular vectors of same magnitude are perpendicular to each other.Now we are to prove that

Magnitude of sum is

________________(1)

Now magnitude of difference is

________________(2)From equation (1) and (2) it is proved that sum and differnce are of same length.Q.13 How the two vectors of the same magnitude have to be oriented if they were to be combined to given a resultant equal to a vector of the same magnirude?Ans. when the angle between two vectors of same magnitude is 1200, the magnitude of the resultant has also same value.If given two vecors and their resultant are oriented in such a way that they form an equalitral triangle, then all of them will be of same magnitude. In such condition the given vectors will be oriented at 1200 as shown in fig.

Q.14 The two vectors to be combined have magnitudes 60N and 35N. Pick the correct answer from those given below and tell why is it the only one of the three that is correct.Ans. Maximum magnitude of resultant of the given vectors when they are parallel is 60+35=95N and minimum magnitude when they are antiparallel is 60-35=25N. So only 70N is the correct answer.Q.15 Suppose the sides of clesed polygon represent vectors arranged head to tail. What is sum of these vectors?Ans. As we know that we draw resultant vector in head to tail rule by joining the tail of the first vector with the head of the last vector. In closed polygon the tail of the first vector and head of the last vector are at same point. Thus resultant will be zero in such condition.Q.16 (i)Two ships X and Y are travelling in dirrerent directions at equal speeds.the direction of motion of X is due nirth but to an observer on Y the apparent derction of motion of \x is north west. The actual diretion of motion of Y as obseved from shore will be.A) east B) west C) south east D) south westAns. As apparent direction of motion of X is north east, it means that ship Y is moving towards west(ii) a horizintal force F is appied to a small object of mass m at rest on an inclined plane inclined at an angle Qto the horizontal.find the resultant force on the mass.Ans. Resolving F and W into rectangular components parallel and perpendicular to inclined plane as shown. Resultant force will be Fcos- mgsin

Q.17 If all the components of vectors and are reversed. How

would this alter the X ?

Ans. The vector product of two vectors will remain unchanged by reversing all components of the vectors.

Irfan Ali Khan Lecturer GC CCW (0345-9998898)

When components of and are reversed they becomes – and

– . So their cross product will be

Hence the proofQ.18 Name the three different conditions that could make

Ans. As

a) If is a null vector then

b) If is a null vector then

c) If angle Then

And For

Q.19 Can a body rotate about center of gravity under the action of its weight?Ans. No a body can not rotate about center of gravity under the action of its weight.As weight of the body acts through center of the gravity. As weight passes through axis of rotation(i.e. center of gravity). so it can not produce torque (rotation) about center of gravity. Hence a body can not rotate about center of gravity under the action of its weight.

CHAP 3 MOTION AND FORCES

Q.1 (a) What is diffrernce between uniform and variable velocity. (b) From the explanation of the variable velocity define accelaration. (c) Give SI units of velocity and accelration.

Ans: (a) Uniform velocity: If a body covers equal displacements in equal intervals of time, the body is said to be moving with uniform velocity.When velocity of a body is uniform, its magnitude and direction do not change. Variable velocity: If a body covers unequal displacements in equal inrevals of time, the body is said to be moving with variable velocity.Velocity may be variable due to change in magnitude or direction or both.(b) Accelaration: When a body is moving with variable velocity then, the change in velocity per unit time is called accelarationIf a body is moving with velocity and after time its velocity becomes then accelaration of the body is

When velocity of body is increasing, accelaration is parallel to velocity and is taken as positive. When velocity of body is decreasing, accelaration is opposite to velocity and is taken as negative (c) SI units of velocity: SI unit of velocity are

Short Questions

SI units of accelaration: SI units of accelaration are

Q.2 An object is thrown vertically upward. Discuss the sign of accelaration due to gravity relative to velocity, while the object is in air.Ans. When an object is thrown vertically upward, during upward motion its velocity goes on decreasing, so accelaration due to gravity is directed opposite to velocity and is taken as negative. During downward motion the velocity starts increasing so accelatation and velocity are in same direction and is taken as positive.Q.3 Can velocity of the body reverse its direction when accelaration of the body is constant. If so, give an example:Ans: Yes, velocity of the body can reverse its direction when accelaration of the body is constant.For example When an object is thrown vertically upward, during upward motion its velocity goes on decreasing, at highest point it becomes zero and body starts falling. Velocity changes its directon, but in whole process accelaration due to gravity remains constant. Q.4 A man standing on the top of the tower throws a ball straight up with initial speed Vi and at the same time another ball throws downward with same speed. Which ball will have larger speed when it strikes the ground. Ignor air friction.

Ans: Both balls will have the same speed on sttiking the ground.We know that when a body is thrown vertically upward with some initial velocity, then it will hit the ground with the same velocity. Thus when the ball thrown straight up with initial velocity Vi it will have the same velocity when it returns back. Hence both balls will have the same speed on striking the ground.

Q.5 Explain the circumstances when the velocity and accelaration

of a car are (i) Parallel (ii) Antiparallel (iii) PerpendicularAns: (i) Parallel: when the velocity of the car is increasing along the straight line. Then and are parallel to each other.

(ii) Antiparallel: when the velocity of the car is decreasing along the straight line. Then and are antiparallel to each other.

(iii) Perpendicular: when car is moving along the circular path, its velocity is perpendicular to accelaration

Q.6 Motion with constant velocity is a special case of motion with constant accelaration. Is this statement true? Explain.Ans. Yes this statement is true. When the body moves with constant velocity, Its accelaration remains zero. As accelaration remains constant so we can say that Motion with constant velocity is a special case of motion with constant accelaration. Q.7 Find change in momentum for an object subjected to a given force for a given time and state law of motion in terms of momentum.

Ans. If a force acts on a body of mass m and its velocity

changes from to in time t, then accelaration of the body is given by

............. (1)

According to Newton’s second law of motion.

Irfan Ali KhanLecturerGC CCW0345-9998898

................(2)

Comparing eq. (1) & (2)

This is newton’s second law of motion in terms of momentum.Statement: Time rate of change of momentum of a body is equal to force applied and change in momentum takes place in the direction of applied force.Q.8 Define impulse and show that how it is related to linear momentum.Ans. Impulse: Impulse is equal to the product of force and time for which the force acts

Mathematically

Relation between linear momentum and impulse:

If a force acts on a body of mass m and its momentum changes

from to then according to newton’s second law of motion

Thus impulse of a body is equal to change in its linear momentum.Q.9 State the law of conservation of linear momentum, pointing out the importance of isolated system.Ans. Law of conservation of linear momentum:Total linear momentum of an isolated system remains constant.Isolated system: A closed system on which no external force acts then such system is called isolated system.Importance: There are many applications like

1. Total momentum of Rocket and its fuel remains constant.2. Total momentum of a bullet and gun remains constant.3. Total momentum of molecules in a closed vessal remains

constant.Q. 10 What is difference between elastic and inelastic collision. Explain how a bouncing ball acts in each case.Ans. Elastic collision: A collision in which total linear momentum and total kinetic energy before and after collision remains unchahged.Inelastic collision: A collision in which kinetic energy is lost and does not remain same.When a ball is dropped from certain height, if it encounters an elastic collision it will rebound to the same hieght as it does not loss any energy.if it encounters an inelastic collision it will not rebound to the same height as some part of its energy is lost during collision.Q. 11 At what point or points in its path does a projectile have minimum speed, maximum speed. Ans. The speed of the projectile is minimum at highest point because vertical component of velocity becomes zero at that point.

The speed of the projectile is maximum at launching and landing point because vertical component of velocity is maximum there.

Irfan Ali Khan Lecturer GC CCW 0345-9998898