FORCE AND MASS - City University Londondanny/1_newton.pdfFORCE AND MASS • Fundamentally, a force (N ... Suppose two astronauts are using jet packs ... 1 pushes in the positive x

1. Newton’s Laws of Motion and their

Applications

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FORCE AND MASS

• Fundamentally, a force (N – Newton) is a push or a pull• When you push or pull something, you exert a force

• Holding something out in your hand exerts a force to oppose the downward pull of gravity

• There are two quantities that characterise a force

• Magnitude and direction• Therefore force is a vector, which is a mathematical

quantity with both magnitude and direction• Generally there are several forces acting on an object

• A book resting on a table experiences a downward force due to gravity and an upward force due to the table

• The total, net , force being exerted is the vector sum of the individual forces acting on an object

• The mass (kg) of an object is a measure of how difficult it is to change its velocity


Applications

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NEWTON’S FIRST LAW OF MOTION

(1)

• When you push a book across a table, it stops moving when you stop pushing it

• But it is wrong to say that a force is needed for an object to move

• A force is required to change an object’s motion

• It is the force of friction between the book and the table that causes the book to come to rest

• You would travel much further sliding on ice where the frictional force is a lot smaller than you would sliding with the same initial speed on sand

• If you were to slide on a frictionless surface, you would keep moving with constant velocity forever

• You would only stop moving if another force was applied to stop you

• Newton’s 1 st Law: An object at rest remains at rest as long as no net force acts on it. An object moving with constant velocity continues to move with the same speed and in the same direction as long as no net force acts on it

• No net force : no force acts on an object, or forces act on an object but they sum to zero


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NEWTON’S FIRST LAW OF MOTION

(2)

• Newton’s 1st Law also known as law of inertia• Inertia literally means “laziness”• An object (matter) is “lazy” because it will not change its

motion unless forced to do so• An object at rest will not start moving on its own

• If an object is already moving with constant velocity, it will not alter its speed or direction unless a force causes the change

• Being at rest and moving at constant speed are equivalent

• A more compact statement of the first law is that if the net force on an object is zero, then its velocity is constant


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NEWTON’S SECOND LAW OF

MOTION (1)

• To hold an object in your hand, you must exert an upward force to oppose gravity

• Suddenly removing your hand causes the object to accelerate downwards as the only force acting on it is gravity

• Unbalanced forces cause accelerations

• Newton’s 2nd law can be demonstrated using a force meter, which contains a spring inside

• The scale gives a reading F exerted by the spring• Two equal weights exert twice the force of one


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MOTION (2)

• A calibrated force meter can be used to perform experiments that demonstrate the 2nd law

• It can be used to accelerate a mass on a “frictionless”air track. If the force is doubled, the acceleration is also doubled – acceleration is proportional to the force

• If the mass of an object is doubled but the force remains the same, the acceleration is halved –acceleration is inversely proportional to mass


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MOTION (3)

• There may be several forces acting on a given mass, and these forces maybe in different directions

• Sum of force vectors =

• Acceleration is also a vector: it has magnitude, direction• Thus Newton’s 2nd Law is:

• In words: If an object of mass m is acted on by a net force , it will experience an acceleration that is equal to the net force divided by the mass. Because the net force is a vector, the acceleration is also a vector. The direction of an object’s acceleration is the same as the direction of the net force acting on it

• Vector components:

• When , then . When acceleration is zero, then the velocity is constant, thus Newton’s first and second laws are consistent with one another.

∑= FFrr

net

aFF

arr

r

rm

m== ∑

∑ or

∑Fr

ar

zzyyxx maFmaFmaF === ∑∑∑ , ,

0=∑Fr

0=ar


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FREE BODY DIAGRAMS

• When solving problems involving forces and Newton’s laws, it is necessary to make a sketch that indicateseach and every external force acting on an object

• This is called a free body diagram• In such diagrams, each object of interest is treated as a

point particle and each of the forces acting on the object are applied to that point


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FREE BODY DIAGRAMS: EXAMPLE

• Mark, Larry and Colin push on a 752kg boat that floats next to a dock. They each exert an 80.5N force parallel to the dock. What is the acceleration of the boat if they all push in the same direction? Give both direction and magnitude. What is the magnitude and direction of the boat’s acceleration if Larry and Colin push in the opposite direction to Mark’s push?


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FREE BODY DIAGRAMS: REAL

WORLD EXAMPLE

• Foamcrete is a substance designed to stop an aeroplane that has run off the end of a runway, without causing injury to passengers. It is solid enough to support a car, but crumbles under the weight of a large aeroplane. By crumbling, it slows the plane to a safe stop. Suppose a 747 with a mass of 1.75×105kg and an initial speed of 26.8m/s is slowed to a stop in 122m. What is the magnitude of the retarding force exerted by the Foamcrete on the airliner?

Fr


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NEWTON’S THIRD LAW OF MOTION

• Forces always come in pairs

• Forces in a pair are equal in magnitude, but opposite in direction

• Newton’s 3rd law in words: For every force that acts on an object, there is a reaction force acting on a different object that is equal in magnitude and opposite in direction

• If object 1 exerts a force on object 2, then object 2 exerts a force on object 1

Fr

Fr

−


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NEWTON’S THIRD LAW OF MOTION:

EXAMPLE

• Two groups of canoeists meet in the middle of a lake. After a brief visit, a person in canoe 1 pushes on canoe 2 with a force of 46N to separate the canoes. If the mass of canoe 1 and its occupants is m1 = 150kg, and the mass of canoe 2 and its occupants is m2 = 250kg, find the acceleration the push gives to each canoe. What is the separation of the canoes after 1.2s of pushing?


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CONTACT FORCES

• When objects are touching one another, the action-reaction forces are often referred to as contact forces

• Example: A box of mass m1 = 10.0kg rests on a smooth, horizontal floor next to a box of mass m2 = 5.00kg. If you push on box 1 with a horizontal force of magnitude F = 20.0N, what is the acceleration of the two boxes? What is the force of contact between the boxes? (The force exerted by one box on the other is different, depending on which one you push – prove this by calculating the contact force when pushing on box 2 with the same force of 20N)


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THE VECTOR NATURE OF FORCES:

FORCES IN 2D

• If more than one force acts on an object, then its acceleration is in the direction of the vector sum of the forces

• Force and acceleration have both magnitude and direction

• Mass is simply a positive number with no direction• Example: Suppose two astronauts are using jet packs

to push a 940kg satellite toward the space shuttle. With the coordinate system indicated in the figure, astronaut 1 pushes in the positive x direction and astronaut 2 pushes in a direction of 52°above the x axis. If astronaut 1 pushes with a force of magnitude F1 = 26N, and astronaut 2 pushes with a force of magnitude F2 = 41N, what are the magnitude and direction of the satellite’s acceleration?


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THE VECTOR NATURE OF FORCES:

FORCES IN 2D: EXAMPLE

• Jack and Jill lift upward on a 1.3kg bucket of water, with Jack exerting a force of magnitude 7.0N and Jill exerting a force of magnitude 11N. Jill’s force is exerted at an angle of 28°with the vertical. At wha t angle θ with respect to the vertical should Jack exert his force if the bucket is to accelerate straight upward?

1Fr

2Fr


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WEIGHT

• When you step onto a scale to weigh yourself, the scale gives a measurement of the pull of Earth’s gravity – this is your weight

• The weight, W, of an object on the Earth’s surface is the gravitational force exerted on it by the Earth

• The greater the mass of an object, the greater its weight

• W = mg; Unit it the newton (N), g = 9.81m/s and is constant for objects falling due to the Earth’s gravity

• Weight, W, and g are both vector quantities• Thus

• Example: A 97kg fireman slides 3.0m down a pole to the ground floor. Suppose the fireman starts from rest, slides with constant acceleration, and reaches the ground in 1.2s. What is the upward force exerted by the pole on the fireman?

gWrr

m=

Fr


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APPARENT WEIGHT

• If you are in a lift moving downward which comes to a rest by accelerating upward, you feel heavier

• You feel lighter when a lift moving upward comes to a rest by accelerating downward

• Motion of lift gives rise to an apparent weight that differs from our own weight

• Consider an elevator that is moving with an upward acceleration a

• Sum of forces acting is

• ΣFy = Wa – W

• By Newton’s 2nd law, sum must equal may (ay = a)

• Wa – W = ma• The apparent weight Wa is

• Wa = W + ma = mg + ma

• If the lift accelerates downward, a is –a, thus

• Wa = W – ma = mg - ma


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APPARENT WEIGHT: EXAMPLE

• As part of an attempt to combine physics and biology in the same class, a lecturer asks students to weigh a 5kg salmon by hanging it from a fish scale to the ceiling of a lift. What is the apparent weight of the salmon, , if the elevator is at rest, moves with an upward acceleration of 2.5m/s2, and moves with a downward acceleration of 3.2m/s2?

aWr


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NORMAL FORCES

• When an object is resting on a surface its acceleration is zero, so the net force acting on it is zero

• Thus the downward force of gravity is being opposed by an upward force exerted by the surface

• This force is called the normal force• The reason why this force is called normal is that it is

perpendicular to the surface. In mathematical terms, normal simply means perpendicular

• In the figure below, the magnitude of the normal force is equal to the weight of the tin

• However, in general the normal force maybe be greater or less than the weight of an object

Nr


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EXAMPLE OF NORMAL FORCE

DIFFERING FROM WEIGHT

• Consider pulling a 12kg suitcase across a smooth floor by exerting a force of 45N at angle 20°above the horizontal

• Weight of suitcase is mg = 12×9.8 = 118N

• The suitcase does not move in the y direction, so its ycomponent of acceleration is 0: ay = 0 → ΣFy = may = 0

• So we can solve for the one force that is unknown

• ΣFy = Wy + Fy + Ny = - mg + Fsin20 + N = 103NNr


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NORMAL FORCES: EXAMPLE

• A 6kg block of ice is acted on by two forces as shown in the diagram. If the magnitude of the forces are F1 = 13N and F2 = 11N, find the acceleration of the ice and the normal force exerted by it on the table.


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FRICTIONAL FORCES

• Previously we had assumed that that surfaces were smooth – objects could slide without resistance to their motion

• No surface is perfectly smooth• Atomically, smooth surfaces are jagged and rough

• To slide one surface across another requires a force large enough to overcome the resistance of microscopic hills and valleys colliding into one another

• This is the origin of the force called friction• Some friction is desirable, such as when walking the

force that propels you is the friction between your shoes and the ground (harder to walk on ice!)

• Two types of friction: kinetic and static


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KINETIC FRICTION

• Kinetic friction is the friction encountered when surfaces slide against one another

• The force generated by this friction is designated with fk• fk acts to oppose sliding motion at the point of contact

• The force of kinetic friction is proportional to the magnitude of the normal force, N

• Thus: fk= µkN → not a vector equation• µk is the coefficient of kinetic friction (dimensionless)

• The greater µk, the greater the friction; the smaller µk, the smaller the friction

• The force of kinetic friction is independent of the relative speed of the surfaces, and the area of contact between the surfaces


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KINETIC FRICTION: EXAMPLE

• A trained sea lion slides from rest with constant acceleration down a 3.0m long ramp into a pool of water. If the ramp is inclined at an angle of 23° above the horizontal, and the coefficient of kinetic friction between the sea lion and the ramp is 0.26, how long does it take for the sea lion to make a splash in the pool?


Applications

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STATIC FRICTION

• Static friction keeps two surfaces from moving relative to one another

• The force of static friction is fs• Consider moving a brick, as shown below• At rest, fs = 0, then fs = F1 but the brick does not move

• When fs = F2, the brick stays at rest, but any bigger than F2, the brick will move

• Thus there is an upper limit that can be exerted by static friction, fs,max → 0 ≤ fs ≤ fs,max

• This maximum is also proportional to magnitude of normal force → , fs,max = µsN (non vector )

• µs is dimensionless and is generally greater than µk

• fs independent of area of contact between surfaces


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STAIC FRICTION: EXAMPLE

• A flatbed truck slowly tilts its bed upward to dispose of a 95kg load. For small angles of tilt, the load stays put, but when the lift angle exceeds 23.2°, the crate begins to slide. What is the coefficient of static friction between the bed of the truck and the load?


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STRINGS AND SPRINGS

• A string pulled from either end has a tension T. If the spring were to be cut at any point, the force required to hold the ends together is T

• In a heavy rope, the tension is noticeably different at points 1, 2 and 3 (higher tension). As the rope becomes lighter, the difference in tension decreases.

• We assume that ropes are massless, hence the tension is uniform

• Pulleys are used to redirect a force exerted by a rope

• An ideal pulley (massless and frictionless) simply changes the direction of the tension in a rope without changing its magnitude


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TENSION: EXAMPLE

• A traction device employing three pulleys is applied to a broken leg as shown. The central pulley is attached to the sole of the foot, and a mass m supplies the tension in the ropes. Find the value of the mass m if the force exerted on the sole of the foot by the central pulley is to be 165N.


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SPRINGS AND HOOKE’S LAW

• A spring exerts a force that is proportional to the amount, x, by which it is stretched or compressed

• F = kx

• k is the force constantor spring constant

• Its units are N/m

• The larger the k, the stiffer the spring

• More precisely, the force exerted by the spring is opposite to the pull (or push)

• Hence Fx = -kx (gives magnitude and direction)

• Ideal springs are massless and obey Hooke’s Law exactly


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TRANSLATIONAL EQUILIBRIUM

• An object is in translational equilibrium when the net force acting on it is 0:

• From Newton’s 2nd law, it is equivalent to saying that the object’s acceleration is zero

• Equilibrium allows us to calculate unknown forces• For the bucket: ΣFy = 0 if lifted with constant speed

• Thus T1 – mg = 0 (W = mg)

• On the pulley: T2 – T1 – T1 = 0 or T2 = 2mg

0=∑Fr


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TRANSLATIONAL EQUILIBRIUM:

EXAMPLE

• To hang a 6.2kg pot of flowers, a gardener uses two wires – one attached horizontally to a wall, the other sloping upward at an angle of θ = 40°and attached to the ceiling. Find the tension in each wire.


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CONNECTED OBJECTS

• If there are two objects connected by a string, and the force and the masses are known, the acceleration and tension can be found

• Each object is considered as a separate system

• Box 1: F – T = m1a Box 2: T = m2a

• Both boxes have the same acceleration• Adding both equations: F = (m1 + m2)a

• Thus a = F/(m1 + m2)

• Finally T = m2a = (m2/m1 + m2)F


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CONNECTED OBJECTS: EXAMPLE

• If there is a pulley, it is easier to have the coordinate system follow the string

• A block of mass m1 slides on a frictionless table. It is connected to a string that passes over a pulley and suspends a mass m2. Find the acceleration of the masses and the tension in the string.


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CIRCULAR MOTION (1)

• When an object moves, a force is required to change its speed, direction or both

• If you drive a car with constant speed on a circular track, the direction of the car’s motion changes continuously

• A force must act on the car to cause this change in direction

• Swinging a ball on the end of a piece of string causes a tension in the string pulling outward

• At the other end of the string where the ball is attached, the tension pulls inward, towards the centre

• To make an object move in a circle with constant speed, a force must act on it that is directed toward the centre of the circle


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CIRCULAR MOTION (2)

• From the previous example of the swinging ball, the ball is acted on by a force toward the centre of the circle

• It must therefore be accelerating toward the centre of the circle

• However, how can a ball moving at constant speed have an acceleration?

• Remember that acceleration is a vector, and is produced whenever speed or direction changes

• In circular motion, the direction changes continuously• This centre-directed acceleration is called centripetal

acceleration• When an object moves in a circle with radius r, constant

speed v, its centripetal acceleration is acp = v2/r

• A force must be applied to an object to give it circular motion

• For an object of mass m, the net force acting on it must have a magnitude given by fcp = macp = mv2/r

• This force is called the centripetal force


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CIRCULAR MOTION: EXAMPLES

• A 1200kg car rounds a corner of radius r = 45m. If the coefficient of static friction between the tires and the road is µs = 0.82, what is the greatest speed the car can have without skidding?

• If a road is banked at the proper angle, a car can round a corner without an assistance from friction between the tires and the road. Find the appropriate banking angle for a 900kg car travelling at 20.5m/s in a turn of radius 85.0m

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FORCE AND MASS - City University Londondanny/1_newton.pdfFORCE AND MASS • Fundamentally, a force (N ... Suppose two astronauts are using jet packs ... 1 pushes in the positive x