A.2 Forces and momentum

A book rests on a horizontal table.

Which force is the Newton's third law pair to the gravitational force exerted on the book by the Earth?

Forces can be classified as contact forces or field forces.

Which row gives a contact force and a field force?

A parachutist is descending vertically at a constant speed.

Which statement about the forces on her is correct?

A wooden block floats at rest on water, partly submerged.

Which statement about the buoyancy force on the block is correct?

A spring extends by 4.0 cm when a force of 12 N is applied to it. The spring obeys Hooke's law.

What is the spring constant?

An object at rest explodes into two fragments that move off in opposite directions. No external force acts.

Which row describes the total momentum and the total kinetic energy of the two fragments just after the explosion?

A block slides down a rough slope. In each diagram the slope descends to the right and the block is moving down the slope.

Which diagram is the free-body diagram of the block?

A van of mass 3600 kg and a car of mass 1200 kg collide head-on.

During the collision, what are (magnitude of force on car) ÷ (magnitude of force on van) and (magnitude of acceleration of car) ÷ (magnitude of acceleration of van)?

A gymnast lands vertically with the same speed on a hard floor and on a thick crash mat, and is brought to rest each time. The time taken to stop is 0.040 s on the floor and 0.32 s on the mat.

What are (change of momentum of the gymnast on the mat) ÷ (change of momentum of the gymnast on the floor) and (average resultant force on the gymnast on the mat) ÷ (average resultant force on the gymnast on the floor)?

A car travels around a horizontal circular bend at constant speed.

Which statement about the resultant force on the car is correct?

A passenger is standing, without holding on, in a bus that is moving at constant velocity along a straight, level road. The bus brakes suddenly and the passenger starts to fall forwards relative to the bus.

Which statement best explains this?

A robotic rover has a mass of 1025 kg on the Earth. The gravitational field strength at the surface of Mars is 3.71 N kg⁻¹.

Which row gives the mass and the weight of the rover on the surface of Mars?

One end of a spring is fixed. The other end is moved along the axis of the spring so that its displacement from the natural length is x, where x is positive for an extension and negative for a compression. The spring obeys Hooke's law for both.

Which graph shows how the elastic restoring force F exerted by the spring varies with x?

The viscous drag force on a small sphere moving through a fluid is F_d = 6πηrv.

What is the unit of the viscosity η in SI base units?

Which row correctly identifies the force that provides the centripetal force?

A warehouse robot pushes horizontally with force F on trolley P, of mass 2m. Trolley P is coupled to trolley Q, of mass m, which is in front of it. Friction is negligible. What is the magnitude of the force in the coupling between P and Q?

A curling stone A slides with speed v and makes a head-on elastic collision with an identical stone B, which is at rest. Friction during the collision is negligible. What happens immediately after the collision?

An astronaut floats at rest outside a space station. Her mass, including her suit but not the tool she is holding, is 90 kg. She throws a 3.0 kg tool away from her. The tool moves at 6.0 m s⁻¹. What is her recoil speed?

A gymnast lands from the same height either on a thick foam mat or on a hard floor. In both cases she comes to rest without bouncing. Compared with landing on the floor, landing on the mat gives

A raindrop and a hailstone have the same mass and both strike a horizontal car roof vertically at the same speed v. The raindrop comes to rest on the roof. The hailstone rebounds vertically with speed v/2.
What is the ratio \(\frac{\text{impulse exerted on roof by hailstone}}{\text{impulse exerted on roof by raindrop}}\)?

A communications satellite is in a circular orbit around the Earth. Earth exerts a gravitational force on the satellite.
According to Newton's third law, what is the other force in this pair?

At an airport, a suitcase is placed at rest on a horizontal conveyor belt that is moving forward at constant speed. For a short time the suitcase slips on the belt until it reaches the belt's speed.
While the suitcase is slipping, what is the direction of the frictional force exerted by the belt on the suitcase?

An astronaut of total mass 80 kg floats at rest just outside a space station. She throws a 2.0 kg wrench directly away from the station at 4.0 m s⁻¹.
What is the velocity of the astronaut immediately afterwards?

A blood sample tube in a laboratory centrifuge moves in a horizontal circle of fixed radius. At one rotation rate, the centripetal acceleration of the tube is 2000 m s⁻². The angular speed of the rotor is then doubled while the radius stays the same. What is the new centripetal acceleration of the tube?

A bird feeder hangs from two identical light springs joined end to end, one below the other. Each spring has spring constant k. What is the spring constant of the combination?

The spring in a luggage balance has a natural length of 20.0 cm. When a 4.0 N load is hung from it, its length becomes 25.0 cm. The spring obeys Hooke's law. What is the length of the spring when a 10.0 N load is hung from it?

A point on the platter of a computer hard-disk drive moves in a circle at a constant rate of rotation. Which quantity for this point changes as it moves?

A delivery drone hovers at rest above a garden. The Earth exerts a gravitational force on the drone. According to Newton's third law, what is the partner of this force?

An astronaut of mass 80 kg floats inside the International Space Station (mass 4.2 × 10⁵ kg). She pushes gently on a wall and moves away from it. During the push, which statement is correct?

A space probe is far from any star or planet, so gravitational forces on it are negligible. It is moving at 15 km s⁻¹ when its engine is switched off. What happens to the probe afterwards?

A magnet is held at rest on the vertical steel door of a refrigerator. Which force prevents the magnet from sliding down?

A smartphone of mass m rests without sliding on the rough surface of a stand tilted at angle θ to the horizontal. The coefficient of static friction between the phone and the stand is μ. What is the magnitude of the frictional force on the phone?

A spring-loaded toy at rest on a frictionless track splits into two parts, X and Y. X has mass m and Y has mass 3m. What is \(\frac{\text{kinetic energy of X}}{\text{kinetic energy of Y}}\) immediately after the split?

A 50 kg crate rests on a level warehouse floor. The coefficient of static friction between the crate and the floor is 0.40. A worker pushes horizontally on the crate with a force of 150 N and the crate does not move. What is the frictional force on the crate?

A tennis ball machine launches balls of mass 58 g horizontally from rest to a speed of 30 m s⁻¹. It fires 2 balls every second. What is the average horizontal force exerted by the machine on the balls?

A rocket engine on a test stand ejects exhaust gas at a rate of 40 kg s⁻¹ with a speed of 2500 m s⁻¹ relative to the engine. What is the thrust on the engine?

A probe in deep space, far from any gravitational influence, fires its engine. The thrust and the rate at which fuel is ejected are both constant. How does the acceleration of the probe change while the engine is firing?

A remotely operated vehicle (ROV) used to inspect an offshore wind-turbine foundation is fully submerged in sea water. It descends from a depth of 20 m to a depth of 200 m. The volume of the ROV and the density of the sea water stay constant. What happens to the buoyant force on the ROV during the descent?

In an environmental-science laboratory, a spherical microplastic bead of radius r moves through still water at speed v and experiences a viscous drag force F given by Stokes's law. A second bead of radius 2r moves through the same water at speed v/2. What is the drag force on the second bead?

A hospital lift carries a patient on a trolley. The lift is moving downward at a constant speed. What is the resultant force on the trolley and patient?

A small ball on a string moves in a vertical circle. Consider the instant when the ball is at the lowest point of the circle.

Which row compares the tension in the string with the weight of the ball, and gives the direction of the resultant force on the ball, at that instant?

A block slides down a rough plane inclined at an angle θ to the horizontal. The coefficient of dynamic friction between the block and the plane is μ_d.

What is the magnitude of the acceleration of the block?

A lamp of weight W hangs at rest from two identical cables. Each cable makes an angle θ with the vertical, as shown.

What is the tension in each cable?

A ball of mass 0.40 kg is at rest on a smooth horizontal surface. It is struck, and the graph shows how the force on it varies with time.

What is the speed of the ball immediately afterwards?

A disc rotates about its axis at a constant rate. Point X on the disc is a distance r from the axis and point Y is a distance 2r from the axis.

What are the ratios of the speed and of the centripetal acceleration of Y to those of X?

A ball of mass m hits a wall at speed u and rebounds along the same line at speed v. The ball is in contact with the wall for a time Δt.

What is the magnitude of the average force exerted on the ball by the wall?

A gardener pushes a lawnmower of mass m across level grass at constant velocity. She pushes along the handle with a force F directed at an angle θ below the horizontal.

Which row gives the horizontal resistive force on the mower and the normal force exerted on the mower by the ground?

A freight wagon P of mass 2.0 × 10⁴ kg moves east at 3.0 m s⁻¹ along a straight, level track. It collides with a wagon Q of mass 1.0 × 10⁴ kg moving west at 4.0 m s⁻¹, and the wagons couple together. External horizontal forces are negligible.

What are the speed and the direction of the coupled wagons just after the collision?

A rocket of total mass M stands on its launch pad. When its engine fires, exhaust gas of mass R is ejected vertically downwards each second at speed u relative to the rocket.

What is the acceleration of the rocket just after it leaves the pad, while its speed is still very small?

A brick of weight W rests, without sliding, on a rough plank that is inclined at an angle θ to the horizontal. Which row gives the magnitude and the direction of the total contact force exerted by the plank on the brick?

A planter of weight W is supported by two identical elastic cords that obey Hooke's law. The masses of the cords are negligible.

When the cords hang side by side, both attached to the planter, the extension of each cord is x_P. When the same cords are joined end to end and the planter hangs from the lower end, the total extension of the two cords is x_S.

What is x_S/x_P?

An electric tug pulls two baggage carts in a straight line along a level floor. Cart P, of mass 2m, is coupled to the tug, and cart Q, of mass m, is coupled behind P. The tug exerts a horizontal force F on P. Resistive forces on the carts are negligible.

What is the magnitude of the force exerted by P on Q?

A small bubble of gas rises vertically at its terminal speed through a glass of sparkling water. The weight of the bubble is W and the buoyancy force on it is F_b.

Which row gives the direction of the viscous drag on the bubble and compares F_b with W?

The graph shows how the momentum p of a trolley moving along a straight track varies with time t.

What is the magnitude of the resultant force on the trolley at t = 3.0 s?

A ball is released from rest and falls freely towards the Earth. Air resistance is negligible.

Which statement about momentum during the fall is correct?

Far from any planet, a capsule of mass 1200 kg and a service module of mass 3600 kg are joined together and at rest. A spring mechanism pushes them apart along a straight line. No external force acts.

What is (kinetic energy of the capsule) ÷ (kinetic energy of the service module) after the separation?

The blades of a wind turbine are 60 m long and rotate at a steady 15 revolutions per minute.

Which row gives the angular velocity of the blades and the speed of a blade tip?

A sample tube in a laboratory centrifuge moves in a horizontal circle of radius r at a steady N revolutions per minute. The tube is moved to a radius 2r and the rate is reduced to N/2 revolutions per minute.

What is (new centripetal acceleration of the tube) ÷ (original centripetal acceleration)?

An aircraft flies horizontally at a steady speed of 250 m s⁻¹. Each of its engines takes in 180 kg of air every second and ejects it backwards at 600 m s⁻¹ relative to the aircraft. The mass of fuel added to the air is negligible.

What is the thrust of one engine?

A supply capsule of mass m approaches a space station of mass 3m with speed v, measured relative to the station. The capsule docks and the two move off together. What fraction of the initial kinetic energy, measured in the station's original rest frame, remains as kinetic energy after docking?

In a model of a lift, a 3.0 kg mass and a 1.0 kg mass hang from opposite ends of a light string passing over a frictionless pulley. The masses are released from rest. What is the tension in the string while the masses move?

A firework shell at rest explodes into two fragments of masses m and 3m, which move in opposite directions. What is the ratio \(\frac{\text{kinetic energy of lighter fragment}}{\text{kinetic energy of heavier fragment}}\)?

A block on a rough horizontal bench is connected by a light string, passing over a frictionless pulley at the bench edge, to a hanging mass m. The hanging mass accelerates downwards. Which statement about the tension T in the string is correct?

A locomotive pulls five identical carriages along a straight track with a constant acceleration. The resistive force on each carriage is the same. What is the ratio \(\frac{\text{tension in coupling between locomotive and carriage 1}}{\text{tension in coupling between carriage 4 and carriage 5}}\)?

A golf ball of mass 0.040 kg is initially at rest on a tee. When the club strikes it, the force on the ball rises uniformly from zero to a maximum of 800 N. The force then falls uniformly back to zero. The total contact time is 8.0 ms.
What is the speed of the ball as it leaves the club?

An ion thruster on a deep-space probe ejects xenon ions at a rate of 2.0×10⁻⁶ kg s⁻¹. The ions leave at a speed of 4.0×10⁴ m s⁻¹ relative to the probe.
What is the thrust on the probe?

A person of mass 70 kg stands on a set of bathroom scales in a lift in a tall office building. The scales read 600 N.
Which describes a possible motion of the lift?

In a railway freight yard, a wagon moving at speed v collides with an identical stationary wagon on a straight horizontal track. The two wagons couple together and move off as one.
What fraction of the initial kinetic energy is transferred to other forms in the collision?

A bobsleigh travels around a banked bend at exactly the speed for which no sideways friction is needed. Air resistance can be ignored. Which statement about the bobsleigh at this instant is correct?

A stunt pilot of mass m flies a vertical loop of radius r. At the lowest point of the loop the aircraft has speed v. What is the magnitude of the force exerted by the seat on the pilot at this point?

An engineer cuts a uniform spring of spring constant k into three equal lengths. She then mounts the three pieces side by side (in parallel) to support a load. What is the spring constant of the new arrangement?

A design for a rotating space habitat uses a ring of radius 39.2 m. Astronauts standing on the inside of the outer wall should have a centripetal acceleration of 9.8 m s⁻². What is the required period of rotation of the ring?

An e-scooter rider turns on a flat, horizontal circular path of fixed radius. On a dry surface the maximum speed without skidding is v. On ice, the maximum frictional force available is one quarter of its dry value. What is the maximum speed on ice?

A hand strap hangs freely from the ceiling of a train that is accelerating uniformly along a straight, level track. The strap makes a steady angle θ with the vertical. What is the magnitude of the acceleration of the train?

A spacecraft in deep space fires its engine. Exhaust gas leaves the nozzle at 2500 m s⁻¹ relative to the spacecraft, at a rate of 40 kg s⁻¹. What is the magnitude of the thrust on the spacecraft?

In a warehouse, a robot pushes horizontally with a force of 24 N on a 6.0 kg box. This box is in contact with a 2.0 kg box in front of it. The floor is frictionless. What is the magnitude of the force that the 6.0 kg box exerts on the 2.0 kg box?

A crate of medical supplies is dropped from an aircraft. Just after its parachute opens, the crate is still moving downward but is slowing down. Which statement describes the forces on the crate at this moment?

Grain falls vertically onto a horizontal conveyor belt at a steady rate of 15 kg s⁻¹. The belt moves at a constant 2.0 m s⁻¹. Ignore friction in the rollers. What horizontal force is needed to keep the belt moving at constant speed?

A parcel sits on the flat floor of a delivery van. The van accelerates forward along a level road and the parcel does not slide relative to the floor. What is the direction of the frictional force that the floor exerts on the parcel?

In a ski-wax test, a ski slides down a snow slope inclined at θ to the horizontal at constant velocity. The same ski is then launched up the same slope. While it moves up the slope, what is the magnitude of its deceleration?

An astronaut of mass 90 kg, including her spacesuit but not the wrench she is holding, floats at rest next to a space station. She throws a 1.5 kg wrench, which moves away at 6.0 m s⁻¹ relative to the station. What is her speed relative to the station immediately afterwards?

A stationary nucleus with nucleon number 226 emits an alpha particle (nucleon number 4). The daughter nucleus has nucleon number 222. Treat the masses as proportional to the nucleon numbers. What fraction of the total kinetic energy released is carried by the alpha particle?

Two aluminium blocks, P of mass m and Q of mass 2m, are placed side by side on a wooden board. One end of the board is slowly raised. Both blocks have the same coefficient of static friction with the board. Which statement is correct?

A satellite's ion thruster ejects xenon ions at a steady rate of 3.0 mg s⁻¹. The ions leave with a speed of 30 km s⁻¹ relative to the satellite. What is the thrust on the satellite?

A horizontal jet of water strikes a flat vertical wall. The water runs down the wall with negligible horizontal velocity, and the force on the wall is F. The same jet is now aimed at a curved surface that sends the water straight back horizontally with its speed unchanged. What is the force on the curved surface?

An open-topped railway wagon rolls freely along a straight, horizontal, frictionless track. Rain falls vertically into the wagon and collects inside it. Air resistance is negligible. Which statement describes the wagon and the water it contains?

A leaf blower forces air out through a nozzle of fixed cross-sectional area. The air leaves at speed v after starting from rest. The fan is adjusted so that the air leaves the same nozzle at speed 2v. By what factor does the reaction force on the blower increase?

A helicopter of mass 1200 kg hovers at rest. Its rotor pushes air that was initially at rest vertically downwards at a speed of 20 m s⁻¹. What is the mass of air pushed downwards each second?

An ion thruster on a satellite produces a thrust of 0.090 N. The ions leave the thruster with a speed of 30 km s⁻¹ relative to the satellite. What is the mass of ions ejected per second?

In a blood sedimentation test, red blood cells are modelled as small spheres falling at terminal speed through plasma, with Stokes's law applying. In some illnesses the cells clump into aggregates. Model each aggregate as a sphere of the same density as a single cell but with 3 times the radius. What is the ratio

terminal speed of an aggregate ÷ terminal speed of a single cell?

A helium weather balloon carrying a radiosonde rises vertically through still air at constant velocity. Which statement about the forces on the balloon and its load is correct?

A floating hydrometer is used to test the electrolyte in a lead–acid battery. It floats upright, first in pure water and then in battery acid, which is denser than water. Compared with floating in water, how do the buoyant force on the hydrometer and the volume of the hydrometer below the surface change when it floats in the acid?

A camera drone of weight 12 N hovers at a fixed position in a steady horizontal wind. The wind exerts a horizontal drag force of 5.0 N on the drone. What is the magnitude of the thrust that the rotors must provide?

A skydiver falls vertically at a terminal speed of 50 m s⁻¹ and then opens her parachute. Immediately after the parachute opens, what are the directions of her velocity and of the resultant force on her?

A crate rests on a rough horizontal floor. A horizontal force P is applied to the crate and is increased slowly from zero. The coefficient of static friction is greater than the coefficient of dynamic friction.

Which graph shows how the frictional force F on the crate varies with P?

Two solid spheres X and Y, made of the same steel, fall vertically through the same oil at their terminal speeds. The radius of Y is twice the radius of X. The viscous drag on a sphere is F_d = 6πηrv.

What is (terminal speed of Y)/(terminal speed of X)?

Sand falls vertically onto a horizontal conveyor belt at a constant rate of R kilograms per second and is carried away by the belt. The belt moves at a constant speed v. Friction in the belt mechanism is negligible.

What are the horizontal force that the motor must exert on the belt and the power the motor must deliver?

A body of mass m moving with speed u collides with a stationary body of mass M. The two bodies stick together. No external force acts.

What fraction of the initial kinetic energy remains as kinetic energy after the collision?

A small bob on a light string moves at constant speed in a horizontal circle of radius r. The string makes a constant angle θ with the vertical, as shown.

What is the speed of the bob?

Trolley X moves towards a stationary trolley Y along a horizontal track. A spring buffer on Y is compressed during the collision, which is elastic. Friction is negligible.

Which statement is correct at the instant when the spring is most compressed?

Trolley X of mass 1.0 kg moves at 4.0 m s⁻¹ along a horizontal track towards a stationary trolley Y of mass 3.0 kg. Friction is negligible. Velocities are positive in the original direction of X.

Which row could give the velocities of X and Y after an elastic collision?

A student holds a book of mass m at rest against a vertical wall by pushing it horizontally towards the wall with a force P. The coefficient of static friction between the book and the wall is μ_s. The book does not slide.

What is the magnitude of the frictional force exerted by the wall on the book?

An aerobatic aircraft flies in a vertical circle of radius r. At the top of the circle the pilot is upside down, and the seat pushes on the pilot with a force equal to three times the pilot's weight.

What is the speed of the aircraft at the top of the circle?