A.2 Forces and momentum

A stationary ball of mass 0.25 kg is struck. The force on the ball rises linearly from zero to 400 N and then falls linearly back to zero, the whole contact lasting 0.020 s.

What is the speed of the ball immediately after the strike?

Two blocks of mass 2.0 kg and 3.0 kg are in contact on a frictionless horizontal surface. A horizontal force of 10 N pushes on the 2.0 kg block so that both blocks accelerate together.

What is the magnitude of the force exerted by the 2.0 kg block on the 3.0 kg block?

A block of mass m is at rest on a rough plane inclined at 30° to the horizontal, as shown.

What is the magnitude of the friction force on the block?

A crate stands on the floor of a lift that is accelerating upwards.

Which force forms a Newton's third law pair with the weight of the crate?

The spring in an angler's spring balance has a length of 0.150 m when unloaded. When a fish of mass 0.60 kg hangs at rest from it, the length of the spring is 0.186 m. g = 9.81 m s⁻².

What is the spring constant of the spring?

A museum conservator hangs a small metal fragment from a newton-meter. The reading is 5.40 N in air and 3.40 N when the fragment is completely submerged in water of density 1000 kg m⁻³.

What is the density of the metal?

A lantern of weight W hangs at rest from the midpoint of a light cable fixed between two posts, as shown. Each half of the cable makes an angle θ with the horizontal.

What is the tension in the cable?

A car travels at constant speed v around a flat horizontal bend that is an arc of a circle of radius r. The frictional force from the road on the tyres towards the centre of the bend is F.

The same car later travels at constant speed 2v around a flat bend of radius 2r. What is the frictional force towards the centre now?

During the spin cycle of a washing machine, a wet towel is pressed against the inside of the vertical cylindrical drum. The towel moves in a horizontal circle at constant speed. Which force provides the centripetal force on the towel?

Spring P has spring constant k. Spring Q has spring constant 2k. Each spring is stretched by the same force F, and both obey Hooke's law. What is the ratio \(\frac{\text{elastic energy stored in P}}{\text{elastic energy stored in Q}}\)?

A communications satellite is in a circular orbit around Earth. Which force forms a Newton's third law pair with the gravitational force exerted by Earth on the satellite?

Two ice-dancers stand at rest on frictionless ice and push each other apart. Dancer X has mass 50 kg and dancer Y has mass 75 kg. Which statement is correct while they are pushing?

A force sensor in a tennis racket records the force on the ball during one impact. The force increases linearly from zero to 400 N in 2.0 ms, then decreases linearly back to zero in a further 3.0 ms. What is the impulse on the ball?

A stationary object explodes into two fragments. One fragment has mass m and the other has mass 3m. What is
\(\frac{\text{kinetic energy of fragment of mass } m}{\text{kinetic energy of fragment of mass } 3m}\)?

In the moderator of a nuclear reactor, a neutron of mass m moving at speed v collides elastically and head-on with a stationary carbon-12 nucleus of mass 12m. The neutron rebounds along its original line with speed 11v/13.

What fraction of the initial kinetic energy of the neutron is transferred to the carbon nucleus?

A small coin rests on a horizontal turntable 0.50 m from the axis. The coefficient of static friction between coin and turntable is 0.40.

What is the greatest angular speed at which the coin remains in place? (g = 9.81 m s⁻²)

A wooden block of mass 1.98 kg hangs at rest from light vertical strings. A pellet of mass 20 g, moving horizontally, hits the block and stays embedded in it. The block and pellet swing together until their centre of mass has risen through a vertical height of 0.20 m. Air resistance is negligible.

What was the speed of the pellet just before it hit the block?

A ball of mass m on a string moves in a vertical circle of radius r. At the top of the circle its speed is v = √(2gr).

What is the tension in the string at the top of the circle?

A crate of mass 12 kg rests on a horizontal floor. The coefficient of static friction is 0.50 and the coefficient of dynamic friction is 0.35. A horizontal pull on the crate is increased slowly from zero. When the crate starts to move, the pull is kept constant at 60 N. g = 9.81 m s⁻².

Which row gives the friction force on the crate when the pull is 40 N, and the acceleration of the crate once it is moving?

In deep space a stationary capsule of mass 5.0 kg is split into three parts by a spring mechanism. A 2.0 kg part moves east at 6.0 m s⁻¹ and a 1.0 kg part moves north at 8.0 m s⁻¹, as shown. The third part has mass 2.0 kg.

Which row gives the speed and the direction of motion of the third part?

An engineer models a seat suspension using three identical springs, each of spring constant k. Two of the springs are placed side by side and support a light plate. The third spring connects this plate to the seat, so it is in series with the pair. What is the effective spring constant of the whole arrangement?

Gravel falls vertically onto a horizontal conveyor belt at a steady rate of 20 kg s⁻¹. The belt moves at a constant speed of 1.5 m s⁻¹. Friction in the rollers is negligible. What additional horizontal force must the motor exert on the belt to keep its speed constant?

A footballer kicks a stationary ball of mass 0.43 kg. A force sensor shows that the force on the ball rises linearly from zero to 1200 N in 3.0 ms. It then falls linearly back to zero over a further 5.0 ms. What is the speed of the ball as it leaves the foot?

A test track has a frictionless banked curve of radius r, designed so that cars travelling at speed v need no sideways friction. A new curve is to be built with the same banking angle, for cars travelling at speed 2v. What radius is required?

A technician holds a 0.50 kg sensor block stationary against a vertical wall by pressing on it with a horizontal force of 20 N. The coefficient of static friction between the block and the wall is 0.60. What is the magnitude of the frictional force on the block?

A raindrop falls vertically through still air. The air resistance on it is proportional to the square of its speed, and its terminal speed is v_t. Buoyancy is negligible. What is the acceleration of the raindrop when its speed is v_t/2?

In a space station, three cargo modules P, Q and R, of masses m, 2m and 3m, float freely in a straight line and in contact. No external forces act on them other than a thruster, which applies a constant force F to P. P pushes Q, which pushes R. What is the ratio \(\frac{\text{force of P on Q}}{\text{force of Q on R}}\)?

A crate rests on the flat bed of a delivery truck. The truck accelerates forwards with acceleration a, and the crate does not slip. The mass of the crate is m and the coefficient of static friction is μ_s. Which describes the frictional force exerted by the truck bed on the crate?

A small ball on a light string moves in a vertical circle. At the top of the circle it has the minimum speed needed to complete the circle. Which statement about the ball at the top is correct?

A roller-coaster car passes through the lowest point of a vertical loop of radius 8.0 m. A passenger feels a normal force from the seat equal to three times her weight. What is the speed of the car at this point?

Two identical balls hit a rigid wall at right angles with the same speed. Ball X sticks to the wall. Ball Y rebounds with the same speed. The contact time of Y with the wall is twice that of X. What is \(\frac{\text{average force on Y}}{\text{average force on X}}\)?

Trolley A moves along a frictionless horizontal track and collides with stationary trolley B. They stick together. Which statements are correct?
I. The total momentum of the two trolleys is conserved.
II. The total kinetic energy of the two trolleys is conserved.
III. The impulse on A has the same magnitude as the impulse on B.

Two gliders, P and Q, each of mass m, are on a horizontal frictionless air track. A light spring of spring constant k is fixed to the front of Q, which is at rest. P moves towards Q with speed u and compresses the spring. The spring obeys Hooke's law.

What is the maximum compression of the spring?

A small ball of mass m is attached to a light string and moves in a vertical circle. At the top of the circle, it has the minimum speed needed to keep the string just taut. Air resistance is negligible. What is the tension in the string when the ball is at the bottom of the circle?

A rocket of total mass 3.0×10⁴ kg stands vertically on its launch pad. At ignition, it ejects exhaust gas at 150 kg s⁻¹, with a speed of 3.0×10³ m s⁻¹ relative to the rocket. What is the initial upward acceleration of the rocket?

Two solid spheres made of the same material, of radii r and 2r, fall through the same viscous oil. Both reach terminal speed, and the drag obeys Stokes' law. What is the ratio \(\frac{\text{terminal speed of larger sphere}}{\text{terminal speed of smaller sphere}}\)?

A 3.0 kg block on a horizontal bench is connected by a light, inextensible cord over a frictionless pulley to a hanging 1.0 kg mass. The coefficient of kinetic friction between the block and the bench is 0.20. The system is released from rest and moves. Take g = 10 m s⁻². What is the tension in the cord?

In a reactor moderator, a neutron of mass m makes a head-on elastic collision with a stationary carbon nucleus of mass 12m. What fraction of the neutron's initial kinetic energy is transferred to the carbon nucleus?

A mass m on a light string moves in a vertical circle. Air resistance is negligible and the string stays taut throughout. What is the tension at the lowest point minus the tension at the highest point?

A student investigates a spring for a toy. Masses m are hung from the spring and its length L is measured with a metre rule; each length reading is ±1 mm. The graph shows the force F = mg against the extension x.

The spring is then used as a mass–spring oscillator. With a 0.300 kg mass attached, the student uses a stopwatch to time 10 vertical oscillations and obtains 5.0 s. The uncertainty in each stopwatch timing, mainly due to reaction time, is ±0.2 s.

A wooden block, loaded with different masses, rests on a horizontal board and is pulled horizontally by a hand-held force sensor. For each normal force F_N the student records the peak reading just before the block starts to slip and the steady reading while it slides at constant speed. Both sets of results are plotted with lines of best fit, which meet the force axis at about 0.2 N instead of at the origin.

A horizontal jet of water leaves a circular nozzle at speed v and strikes a flat vertical plate mounted on a force sensor. The water spreads out sideways over the plate, losing all of its horizontal velocity. The speed is changed with a valve and found from the flow rate; each speed is ±0.1 m s⁻¹. The density of water ρ is 1000 kg m⁻³.

The rate of change of momentum of the water gives a force on the plate F = ρAv², where A is the cross-sectional area of the jet.

A permanent magnet rests on an electronic top-pan balance that reads to ±0.01 g. A stiff horizontal copper wire, held by a clamp that does not touch the magnet or the balance, passes between the poles of the magnet at right angles to its uniform field. The length of wire in the field is 4.0 cm. When there is a current I in the wire, the magnetic force on the wire is vertically upwards.

The balance is not set to zero before the readings are taken. The table and graph show the balance reading for each current.

Astronomers track five large moons of Uranus over several weeks. For each moon the orbital radius r is found from its greatest angular distance from the planet and the known distance of Uranus from the Earth, and the orbital period T is found by timing repeated positions. The orbits are assumed to be circular, with the centripetal force provided by the gravitational force of Uranus, mass M.

The graph shows T² against r³.

A trolley of mass 0.500 kg runs along a level track and collides head-on with a spring buffer mounted on a force sensor fixed to the end of the track. A motion sensor shows that the trolley moves at 0.80 m s⁻¹ towards the buffer before the collision and at 0.52 m s⁻¹ away from it afterwards; each speed is ±0.01 m s⁻¹. The force sensor was not set to zero before the run.

The graph shows the reading of the force sensor against time t. Friction is negligible.

Steel ball bearings of radius r are released at the surface of a tall cylinder of glycerine. Their terminal speeds v are found by timing each ball between two marks low on the cylinder. The density of steel is 7800 kg m⁻³ and that of glycerine is 1260 kg m⁻³.

Balancing weight, buoyancy F_b = ρ_fVg and viscous drag F_d = 6πηrv gives v = 2r²(ρ_s − ρ_f)g/(9η), where η is the viscosity of glycerine. The graph shows v against r².

A rubber bung of mass m is tied to a string that passes through a smooth vertical tube held in the hand. A mass M = 0.200 kg hangs from the lower end of the string, so the tension in the string is Mg. The bung is whirled in a horizontal circle of radius r, kept constant using a marker on the string. The time for 20 revolutions is measured with a stopwatch; each timing has an uncertainty of ±0.3 s.

If the tension provides the centripetal force, T² = (4π²m/Mg)r, where T is the period. The graph shows T² against r.

A small cylindrical magnet falls down the centre of a long vertical copper tube. Brass (non-magnetic) discs are fixed to the magnet to change the total mass m. Two thin coils wound round the tube 0.400 m apart, well below the top, are connected to a data logger; each coil gives a voltage pulse as the magnet passes. The time t between the pulses has an uncertainty of ±0.05 s. The magnet falls at its terminal speed v between the coils.

The resistive force on the magnet is modelled as bv, where b is a constant.

A goods lift and its load have a total mass of 1200 kg. The graph shows how the upward velocity v of the lift varies with time t during one journey.

A delivery drone of mass 4.0 kg carries a parcel of mass 1.5 kg that hangs below it on a light vertical cable. The drone takes off vertically from rest. Air resistance is negligible throughout.

A digital luggage scale contains a steel spring. A student hangs known loads from the spring and measures its extension.

Load / N: 0, 5.0, 10.0, 15.0, 20.0
Extension / cm: 0, 2.0, 4.0, 6.0, 8.0

The manufacturer states that the spring remains within its limit of proportionality for all loads up to 50 N.

A sprinter of mass 68 kg starts a race from rest in starting blocks. For 0.32 s the blocks exert an average horizontal force of 820 N on the sprinter. Ignore all other horizontal forces.

A footballer kicks a stationary ball of mass 0.45 kg. A force plate inside a test boot records the force on the ball. Assume the force varies linearly between readings.

t / ms: 0, 2.0, 4.0, 6.0, 8.0, 10.0
F / N: 0, 300, 600, 600, 300, 0

The ball leaves the boot at t = 10.0 ms.

In a railway yard, wagon A of mass 1.2 × 10⁴ kg moving at 1.50 m s⁻¹ runs into wagon B of mass 8.0 × 10³ kg, which is at rest. The buffers of the wagons are springs with hydraulic dampers. The graph shows how the force exerted by A on B varies with time during the contact. Friction from the track is negligible.

A rope swing hangs from a branch above a river. The rope is 8.0 m long. A person of mass 60 kg holds the end of the rope and steps off a bank, starting from rest with the rope taut at 40° to the vertical. Treat the person as a point mass; air resistance and the mass of the rope are negligible.

Two gliders, A of mass 0.40 kg and B of mass 0.60 kg, rest on a horizontal air track with a light spring of spring constant 800 N m⁻¹ compressed by 2.5 cm between them. A thread holds them together. When the thread is burned, the spring pushes the gliders apart and falls away. Friction is negligible.

In a fairground swing ride, each seat hangs from a light chain of length 4.0 m attached to the rotating top of the ride 3.0 m from its vertical axis. When the top rotates at a steady rate, each chain makes an angle of 40° with the vertical and each seat moves in a horizontal circle, as shown. A seat and its rider have a total mass of 80 kg and may be treated as a point mass at the lower end of the chain.

A medical laboratory centrifuge separates blood samples. It spins at 3000 revolutions per minute. The sample tubes are held so that each sample moves in a horizontal circle of radius 0.12 m about the axis.

A firefighter holds a hose nozzle horizontally. Water of density 1000 kg m⁻³ leaves the nozzle at a speed of 25 m s⁻¹, with a mass flow rate of 12 kg s⁻¹. Assume the water speed inside the hose, just before the nozzle, is negligible.

A delivery drone of mass 2.4 kg carries a package of mass 0.80 kg. The package hangs below the drone on a light vertical cable. The rotors provide a constant upward thrust of 40 N. Air resistance is negligible. Take g = 9.8 m s⁻².

A student places a coin on a horizontal turntable at a distance r from the axis. The student slowly increases the angular speed until the coin begins to slide, and records the angular speed ω at that moment. This is repeated for several values of r.

r / m: 0.05, 0.10, 0.15, 0.20
ω at slipping / rad s⁻¹: 11.1, 7.8, 6.4, 5.5

A hospital bed of total mass 120 kg has its wheels locked, so it can only slide on the floor. The coefficient of static friction between the rubber wheels and the floor is 0.70, and the coefficient of kinetic friction is 0.55. A nurse pushes horizontally on the bed.

A delivery drone releases a parcel attached to a small parachute. Wind-tunnel tests give the drag force F on the parcel and parachute at different air speeds v:

v / m s⁻¹: 2.0, 4.0, 6.0, 8.0
F / N: 0.50, 2.0, 4.5, 8.0

The parcel and parachute have a total mass of 2.4 kg. Buoyancy is negligible. Take g = 9.81 m s⁻².

An electric locomotive of mass 60 × 10³ kg pulls two wagons along a straight, level track. Wagon A (mass 25 × 10³ kg) is coupled directly to the locomotive, and wagon B (mass 15 × 10³ kg) is coupled behind A. The track exerts a forward driving force of 48 kN on the locomotive. Each vehicle experiences a resistive force equal to 0.010 times its own weight.

In a tug-of-war, team A (total mass 420 kg) pulls against team B (total mass 380 kg) using a light, horizontal rope. The coefficients of friction between team B's shoes and the ground are μ_s = 0.80 and μ_k = 0.60. Take g = 9.8 m s⁻².

A satellite of mass 150 kg is released from an upper rocket stage of mass 450 kg (not including the satellite) in deep space. Before release, the two are at rest relative to an inertial observer moving with them. Compressed springs push the satellite away. After release, the satellite moves at 0.60 m s⁻¹ relative to this observer. Assume all the elastic potential energy stored in the springs is transferred to the kinetic energy of the satellite and the stage.

A jet aircraft flies a vertical loop of radius 600 m. The pilot has a mass of 75 kg. At the lowest point of the loop the aircraft's speed is 140 m s⁻¹.

In a crash test, a car of mass 1200 kg moving at 15 m s⁻¹ hits a stationary van of mass 1800 kg. The vehicles lock together. Friction with the road is negligible during the collision.