Forces & Momentum (HL IB Physics)

Describe the microscopic origin of static friction between two objects.

Compare and contrast the static force of friction and the dynamic force of friction.

A block of mass 2.5 kg is at rest on a rough inclined plane. The block just begins to slip down the plane when the angle of inclination is 35°. 

Calculate the coefficient of static friction between the block and the inclined plane.

You may use the following result: 

The angle of inclination is increased to 40°.

Calculate the force that must be applied to the block to move it up the plane with an acceleration of 1.5 m s^–2. 

Use the following data:

• μ_d = 0.30

A ball travels in a circular path on the inside surface of a bowl. 

The normal reaction force N makes an angle φ to the plane of the ball's path. Ignore the effects of friction. 

On the free-body diagram below, construct an arrow to represent the weight of the ball. 

Determine an equation for the resultant force acting on the ball in terms of its mass m and the angle to its plane of orbit φ. 

You may use the result: 

The radius of the ball's orbit decreases.

Explain how the effects of friction are related to the decreasing orbital radius. 

Outline if the ball could travel along a horizontal circular path with an orbital radius equal to the maximum radius of the bowl. Ignore the effects of friction in your answer. 

A woman stands in an elevator and measures the weight of a fish attached to a spring scale. The scale reads 40 N when the elevator is stationary.

Sketch the reading on the spring scale as the lift gently accelerates upward. 

As the elevator continues moving upwards, it gently decelerates to a standstill.

Draw and label a free-body force diagram for the fish as the elevator gently decelerates.

The rope that attaches the spring scale to the ceiling of the elevator suddenly snaps. The spring scale and the fish are momentarily in free-fall – but the observer manages to take a reading from the scale it falls. 

State and explain the reading on the spring scale as it falls. 

Sketch the variation of the contact force on the observer's feet as the elevator decelerates to rest on the axes provided.

The magnitude of the observers weight, mg, is included as a dashed line, for reference.

Two blocks of mass 4.5 kg and 6.0 kg are joined by a string and rest on a smooth horizontal table. A force F of 100 N is applied to one of the blocks. 

Calculate

The 4.5 kg block is now placed on top of the 6.0 kg block. The coeffecient of static friction between them is 0.40. 

Calculate the maximum horizontal force F that can be applied to the bottom block that would result in both blocks moving together without slipping. 

The two blocks are now attached by a light inextensible string that passes over a smooth pulley. They are held stationary and suddenly released. 

Determine the acceleration of each block and the tension in the string.  

The string attaching the blocks over the pulley in part (c) suddenly snaps. 

Describe and explain the subsequent motion of the block of mass 4.5 kg. (Assume it was moving upward at the instant the string snapped). 

Air enters a cargo plane's engine at A and is heated before leaving at B, at a much higher speed.

In one second a mass of 3.75 × 10⁵ g of air enters A and the speed of this mass of air increases by 587 m s⁻¹ as it passes through the engine. 

Calculate the force exerted by the air on the engine.

Hot air flows out of the exhaust engine at B through a cross-sectional area of 5.9 × 10⁶ mm². The density of the hot air is 457.9 g m⁻³. 

Calculate the volume of air leaving the engine every second.

Explain, referring to the momentum of the air as it passes through the engine and using appropriate laws of motion, why it exerts a force on the engine in a forward direction. 

When a cargo plane lands its engines exert a decelerating force on the aircraft by making use of deflector plates. These cause the air leaving the engines to be deflected at an angle to the direction the aircraft is travelling. 

The speed of the deflected air is the same as the speed of the air leaving B. 

A squash ball is raised from the ground and dropped onto a hard plate to test its properties. A sensor measured the force exerted by the plate on the ball during its collision with the plate. 

The variation of force exerted on the squash ball with time is shown on the graph. 

The ball strikes the plate with a speed of 19.16 m s⁻¹ and has a mass of 60 g. It leaves the plate with a speed of 13.83 m s⁻¹.

Show that this is consistent with the impulse obtained from the graph.

Considering the first 30 ms of the ball's motion:

The ball continues to bounce, each time losing the same fraction of its energy when it strikes the plate. Air resistance is negligible. 

Determine the percentage of the original gravitational potential energy of the ball that remains when it reaches its maximum height after bouncing five times. 

Explain, with reference to the conservation of momentum, the effect that the motion of the squash ball has on the motion of the Earth from the instant it is released until it bounces off the plate.

A stationary uranium nucleus decays by emitting an α particle and thorium nucleus. 

Discuss how the principle of conservation of momentum applies in this explosion. 

Assume that all of the energy released in the emission process is transferred as kinetic energy to the α particle and the recoiling nucleus and is equal to 6.46 MeV.

Calculate the kinetic energy of the Thorium nucleus and α particle in MeV. 

Show that momentum is conserved in this decay.

Collisions can occur between neutrons and stationary Uranium nuclei, for example, during nuclear fission.

Discuss how this collision would be if this was inelastic as opposed to elastic.

Hemp ropes were the first to be used for ships' rigging to adjust the position of the sails and support the masts. Hemp is one of the strongest natural fibres available, has a very high tensile strength and is extremely rigid.

Today rock climbers use nylon ropes which are attached to their harness to break their fall. Nylon stretches considerably under tension.

Discuss in terms of impulse and momentum, why nylon ropes are favoured by rock climbers compared to hemp ropes.

A rock climber of mass m slips just after they attach a bolt to a piece of rock as they are climbing.

The length of the rope between their harness and the bolt on the rock face is h. Once the rope is fully extended their speed just before they come to a stop is v. 

Show that the magnitude of the average decelerating force F on the climber when they are brought to rest is equal to their weight. 

Deduce, without calculation, whether the average decelerating force F would still be equal to the climber’s weight if they fell from higher up.

The speed of a dart pellet of mass 2.73 g is measured by firing it into a polystyrene block of mass 543 g suspended from a rigid support. The pellet becomes completely embedded in the polystyrene block. The block can swing freely at the end of a light inextensible string of length 1.5 m measured from the pivot to the centre of the block. 

The centre of mass of the block rises by h at an angle of 35º to the vertical. 

Determine the speed of the pellet when it strikes the polystyrene block. 

The polystyrene block is replaced by a wooden block of the same mass. The experiment is repeated with the wooden block and an identical pellet. The pellet rebounds after striking the block.

A student makes an assumption that the angle that the wooden block makes with the vertical will be greater than 35º because the block doesn’t have the additional mass of the pellet embedded within it.

Discuss the validity of the student's assumptions.

A popular demonstration of the conservation of momentum and conservation of energy is Newton’s cradle. It features several identical polished steel balls hung in a straight line in contact with each other.

If one ball is pulled back and allowed to strike the line, one ball is released from the other end whilst the rest are stationary. If two are pulled out, two are released on the other end and so forth.

Assuming that Newton's Cradle is in a vacuum and considering energy and momentum conservation, explain why swinging one ball from the left will not release two balls on the right. 

A proton of mass m moves with uniform circular motion. Its kinetic energy is K and its orbital period is T .

Show that the orbital radius r is given by: 

The proton moves in a clockwise circle of circumference 1.25 mm. The net force on the proton is 65 fN. 

Determine the linear speed of the proton.

Calculate the proton's orbital frequency. 

A small ball is attached to a string and moves in a horizontal circular path. It completes one revolution every 2.5 s, with the string at an angle θ to the vertical.

Calculate the orbital radius r if θ = 12°. 

You may wish to use the following data: 

Show that the length of the string l is given by: 

You may wish to use the following data: 

The equation in part (b) seems to suggest that the length of the string l is dependent on the angle it makes to the vertical, θ.

Comment on the relationship between the length of the string l and the angle it makes to the vertical, θ. 

A marble rolls from the top of a bowling ball of radius R. 

Show that when the marble has moved so that the line joining it to the centre of the sphere subtends an angle of θ to the vertical, its speed v is given by: 

Deduce that, at the instant shown in the image in part (a), the normal reaction force N on the marble from the bowling ball is given by:

Hence, determine the angle θ at which the marble loses contact with the bowling ball. 

The 'loop-the-loop' is a popular ride at amusement parks, involving passengers in cars travelling in a vertical circle. 

The loop has a radius of 8.0 m and a passenger of mass 70 kg travels at 10 m s^–1 when at the highest point of the loop. 

Calculate, at the highest point:

Stating any assumptions required, calculate the speed of the passenger at the point marked 'exit from loop' in part (a). 

Operators must ensure that the speed of the vehicle carrying passengers into the loop-the-loop is above a certain value. 

Suggest a reason for this, and determine the minimum required speed. 

A popular trick to impress young observers is to swing a bucket of water in a vertical circle. If the bucket is swung fast enough, no water spills out. 

Estimate the minimum linear speed v required to swing a bucket in a vertical circle, such that no water spills. 

Ric is standing in an elevator and has a mass of 68 kg. The elevator is moving upwards at a speed of 2.5 m s^–1 and comes to rest 3.0 s later. 

Calculate the reaction force on Ric from the elevator floor during the time it takes the elevator to come to rest.

Ric realises that he forgot his lab coat on the floor below. He again travels in the elevator which goes downwards at the same speed, coming to rest in the same time.    

Calculate the reaction force on Ric from the elevator floor during the time it takes to come to rest at the floor below.

Hence, deduce which deceleration period of the elevator would make Ric feel:

In a serious stroke of bad luck, the elevator cable snaps and it falls freely. Ric experiences a sense of weightlessness. 

Explain, using appropriate laws of motion, why Ric feels a sense of weightlessness in this situation.

Two blocks A and B are joined by a string and rest on a frictionless horizontal table. A force of 200N is applied horizontally on block B.

Draw free–body diagrams for both boxes.

Block A has a mass of 3.0 kg and block B has a mass of 7.0 kg. 

Calculate the acceleration of each block and the tension in the string.

Block A is now placed on top of block B. The coefficient of static friction between the two blocks is 0.31. The bottom block is pulled with a horizontal force F.

Draw free–body diagrams for both boxes.

Calculate the magnitude of the maximum force F that will result in both blocks moving together without slipping.

A mass is hung from two horizontal strings.

Outline why the mass will not remain in equilibrium when attached to the strings in this way.

A flower pot of mass 15 kg hangs from two strings attached to the ceiling with tensions A and B, where it remains in equilibrium.

Derive two equations for A in terms of B. Give any values to an appropriate number of significant figures.

Hence, calculate tensions A and B.

A gardening enthusiast wants to hang a flower pot from two strings on the ceiling of their balcony. They have two pieces of string, but they are unfortunately quite old and frayed. 

Suggest and explain how they can attach the strings to the flower pot in order for them to stay intact.

A stone block is pulled at constant speed up an incline by a cable attached to an electric motor.

The incline makes an angle of 17º with the horizontal. The mass of the block is 180 kg and the tension T in the cable of 0.9 kN.

On the diagram draw and label arrows that represent the forces acting on the block.

State the type of friction in this system and calculate the frictional force.      

Hence, calculate the appropriate coefficient of friction.

The cable connecting the block to the electric motor abruptly breaks. 

Calculate the acceleration of the block.

Show, using the equation F = ma, how the impulse of a force F is related to the change in momentum Δp that it produces for a mass m with acceleration a.    

A railway truck on a level, straight track is initially at rest. The truck is given a quick, horizontal push by an engine so that it now rolls along the track.

The engine is in contact with the truck for a time T = 0.60 s and the initial speed of the truck after the push is 5.5 m s^–1. The mass of the truck is 3.1 × 10³ kg.

Due to the push, a force of magnitude F is exerted by the engine on the truck. The sketch shows how F varies with contact time t.

Determine the magnitude of the maximum force  exerted by the engine on the truck.

When the speed of the truck is 2.3 m s^–1, it collides with a stationary truck of mass 4.7 × 10³ kg. The two trucks move off together with a speed V.

Show that the speed V = 0.9 m s^–1

State and explain whether the collision of the two trucks is elastic or inelastic.

Two identical blocks A and B of mass 200 g are travelling towards each other along a straight line through their centre. Assume that the surface is frictionless.

Both blocks are moving at a speed of 0.21 m s^–1 relative to the surface.

As a result of the collision, the blocks reverse their direction of motion and travel at the same speed as each other. During the collision, 30% of the kinetic energy of the blocks is given off as thermal energy to the surroundings.

Deduce whether the collision is elastic or inelastic and state your reasoning.

Calculate the final speed of the blocks relative to the surface.

The duration of the collision between the blocks is 650 ms.

Determine the average force one block exerted on the other.

The force acts on a mass of 5.0 kg initially at rest.

Show that the speed of the mass at t = 3 s is 9.0 m s^–1.

Calculate the deceleration of the mass up to time t = 4 s.

Calculate the total impulse experienced by the mass.

Outline the motion of the mass as indicated by the graph.

A space rocket is moving with constant velocity. The engines of the space rocket are turned on and it accelerates by burning fuel and ejecting gases.

Discuss how the law of conservation of momentum allows the space rocket to accelerate forward, although it ejects gases in the opposite direction.

A rocket is travelling at constant velocity in space after exiting the Earth’s atmosphere. The engines are turned off, and a module separates from the rocket.

The module has a mass of 6 000 kg and is ejected at 10 km s^–1. The combined mass of the rocket and the module is 81 000 kg and the remaining part of the rocket after the explosion travels at 4500 m s^–1 after the module has been ejected.

Calculate the initial speed of the rocket.

Calculate the force exerted on the module in 0.2 s during the explosion.

Inside the rocket, some walls are padded to reduce damage to its interior when it is accelerated into space.

Explain, with reference to change in momentum, why padded walls are less likely to cause damage to the interior of the rocket compared to a rigid wall.

Joanna and Lindsay are two roller skaters initially at rest on a horizontal surface. They are facing each other and Joanna is holding a ball. Joanna throws the ball to Lindsay who catches it. The speed at which the ball leaves Joanna, measured relative to the ground, is 6.2 m s^–1.

The following data are available.

Mass of Joanna = 59 kg

Mass of Lindsay = 64 kg

Mass of ball = 3.3 kg

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Calculate the speed V of Lindsay relative to the ground immediately after she catches the ball. Assume the speed of the ball stays constant throughout its motion.

Determine whether Lindsay catching the ball is an elastic or inelastic collision.

Lindsay has a previous injury to her hand, so decides to wear padded gloves whilst playing this game with Joanna. This is similar to what players would wear in cricket if they need to catch a ball at high speed.

Explain why the padded gloves would protect Lindsay’s hands when she catches the ball.

A lead ball of mass 0.55 kg is swung round on the end of a string so that the ball moves in a horizontal circle of radius 1.5 m. The ball travels at a constant speed of 6.2 m s^–1.    

Calculate the time taken for the string to turn through an angle of 170º.

Calculate the tension in the string.

The string will break when the tension exceeds a maximum tension. The ball makes three revolutions per second at the maximum tension of the string.

Calculate the tension above which the string will break.

Describe, using just one of Newton’s Laws, why the ball is accelerating even when its angular speed is constant.

You may wish to draw a diagram to clarify your answer.

Explain why a particle moving in a circle with uniform speed is accelerating.

The diagram shows a fairground ride called a Rotor. Riders stand on a wooden floor and lean against the cylindrical wall.

The fairground ride is then rotated. When the ride is rotating sufficiently quickly the wooden floor is lowered. The riders remain pinned to the wall by the effects of the motion. When the speed of rotation is reduced, the riders slide down the wall and land on the floor.

At the instant shown in below the ride is rotating quickly enough to hold a rider at a constant height when the floor has been lowered.

Explain why the riders slide down the wall as the ride slows down.

A rotor accelerates uniformly from rest to maximum angular velocity of 3.6 rads s^–1. At the maximum speed the centripetal acceleration is 35 m s^–2.  

Calculate the diameter of the rotor.

The diagram shows the final section of a roller coaster which ends in a vertical loop. Cars on the roller coaster descend to the start of the loop and then travel around it.

As the passengers move around the circle from A to B to C, the reaction force between exerted by their seat varies.

State the position at which this force will be a maximum and the position at which it will be a minimum. Explain your answers.