Past Exam Questions

A block is pulled in a straight line along horizontal ground by a force of constant magnitude acting at an angle of 60° above the horizontal. The work done by the force in moving the block a distance of 5 m is 75 J. Find the magnitude of the force.

One end of a light inextensible string is attached to a block. The string is used to pull the block along a horizontal surface with a speed of 2 m s^-1. The string makes an angle of 20° with the horizontal and the tension in the string is 25 N. Find the work done by the tension in a period of 8 seconds.

A block is pulled for a distance of 50 m along a horizontal floor, by a rope that is inclined at an angle of \(\alpha^\circ\) to the floor. The tension in the rope is 180 N and the work done by the tension is 8200 J. Find the value of \(\alpha\).

A particle of mass 1.6 kg is dropped from a height of 9 m above horizontal ground. The speed of the particle at the instant before hitting the ground is 12 m/s^-1.

Find the work done against air resistance.

A particle of mass 0.8 kg slides down a rough inclined plane along a line of greatest slope AB. The distance AB is 8 m. The particle starts at A with speed 3 m/s and moves with constant acceleration 2.5 m/s².

1. Find the speed of the particle at the instant it reaches B.
2. Given that the work done against the frictional force as the particle moves from A to B is 7 J, find the angle of inclination of the plane.
3. When the particle is at the point X its speed is the same as the average speed for the motion from A to B. Find the work done by the frictional force for the particle’s motion from A to X.

A load of mass 160 kg is lifted vertically by a crane, with constant acceleration. The load starts from rest at the point O. After 7 s, it passes through the point A with speed 0.5 m s^-1. By considering energy, find the work done by the crane in moving the load from O to A.

A particle P of mass 0.6 kg is projected vertically upwards with speed 5.2 m/s^-1 from a point O which is 6.2 m above the ground. Air resistance acts on P so that its deceleration is 10.4 m/s^-2 when P is moving upwards, and its acceleration is 9.6 m/s^-2 when P is moving downwards. Find

1. the greatest height above the ground reached by P,
2. the speed with which P reaches the ground,
3. the total work done on P by the air resistance.

A small block is pulled along a rough horizontal floor at a constant speed of 1.5 m s^-1 by a constant force of magnitude 30 N acting at an angle of \(\theta^\circ\) upwards from the horizontal. Given that the work done by the force in 20 s is 720 J, calculate the value of \(\theta\).

A crate of mass 3 kg is pulled at constant speed along a horizontal floor. The pulling force has magnitude 25 N and acts at an angle of 15° to the horizontal, as shown in the diagram. Find

1. the work done by the pulling force in moving the crate a distance of 2 m,
2. the normal component of the contact force on the crate.

A cyclist is riding a bicycle along a straight horizontal road AB of length 50 m. The cyclist starts from rest at A and reaches a speed of 6 m s^-1 at B. The cyclist produces a constant driving force of magnitude 100 N. There is a resistance force, and the work done against the resistance force from A to B is 3560 J.

Find the total mass of the cyclist and bicycle.

A winch operates by means of a force applied by a rope. The winch is used to pull a load of mass 50 kg up a line of greatest slope of a plane inclined at 60° to the horizontal. The winch pulls the load a distance of 5 m up the plane at constant speed. There is a constant resistance to motion of 100 N.

Find the work done by the winch.

A particle of mass 13 kg is on a rough plane inclined at an angle of \(\theta\) to the horizontal, where \(\tan \theta = \frac{5}{12}\). The coefficient of friction between the particle and the plane is 0.3. A force of magnitude \(T\) N, acting parallel to a line of greatest slope, moves the particle a distance of 2.5 m up the plane at a constant speed. Find the work done by this force.

A particle of mass 1.3 kg rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{12}{5}\). The coefficient of friction between the particle and the plane is \(\mu\).

(i) A force of magnitude 20 N parallel to a line of greatest slope of the plane is applied to the particle and the particle is on the point of moving up the plane. Show that \(\mu = 1.6\). [4]

The force of magnitude 20 N is now removed.

(ii) Find the acceleration of the particle. [2]

(iii) Find the work done against friction during the first 2 s of motion. [3]

A particle of mass 1.2 kg moves in a straight line AB. It is projected with speed 7.5 m s^-1 from A towards B and experiences a resistance force. The work done against this resistance force in moving from A to B is 25 J.

1. Given that AB is horizontal, find the speed of the particle at B.
2. It is given instead that AB is inclined at 30° below the horizontal and that the speed of the particle at B is 9 m s^-1. The work done against the resistance force remains the same. Find the distance AB.

A van of mass 2500 kg descends a hill of length 0.4 km inclined at 4° to the horizontal. There is a constant resistance to motion of 600 N and the speed of the van increases from 20 m/s to 30 m/s as it descends the hill. Find the work done by the van’s engine as it descends the hill.

A railway engine of mass 120000 kg is towing a coach of mass 60000 kg up a straight track inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.02\). There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000 N and there are constant resistances to motion of 22000 N on the engine and 13000 N on the coach.

(a) Find the acceleration of the engine and find the tension in the coupling.

At an instant when the engine is travelling at 30 m/s, it comes to a section of track inclined upwards at an angle \(\beta\) to the horizontal. The power produced by the engine is now 4500000 W and, as a result, the engine maintains a constant speed.

(b) Assuming that the resistance forces remain unchanged, find the value of \(\beta\).

A car of mass 1400 kg is moving on a straight road against a constant force of 1250 N resisting the motion.

(a) The car moves along a horizontal section of the road at a constant speed of 36 m s^-1.

1. Calculate the work done against the resisting force during the first 8 seconds.
2. Calculate, in kW, the power developed by the engine of the car.
3. Given that this power is suddenly increased by 12 kW, find the instantaneous acceleration of the car.

(b) The car now travels at a constant speed of 32 m s^-1 up a section of the road inclined at θ° to the horizontal, with the engine working at 64 kW.

Find the value of θ.

A cyclist is travelling along a straight horizontal road. She is working at a constant rate of 150 W. At an instant when her speed is 4 m s^-1, her acceleration is 0.25 m s^-2. The resistance to motion is 20 N.

(a) Find the total mass of the cyclist and her bicycle.

The cyclist comes to a straight hill inclined at an angle \(\theta\) above the horizontal. She ascends the hill at constant speed 3 m s^-1. She continues to work at the same rate as before and the resistance force is unchanged.

(b) Find the value of \(\theta\).

A car of mass 1400 kg is travelling at constant speed up a straight hill inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). There is a constant resistance force of magnitude 600 N. The power of the car’s engine is 22 500 W.

(a) Show that the speed of the car is 11.25 m s\(^{-1}\).

The car, moving with speed 11.25 m s\(^{-1}\), comes to a section of the hill which is inclined at 2° to the horizontal.

(b) Given that the power and resistance force do not change, find the initial acceleration of the car up this section of the hill.

A car of mass 1600 kg is pulling a caravan of mass 800 kg. The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 400 N and 250 N respectively.

(a) The car and caravan are travelling along a straight horizontal road.

1. Given that the car and caravan have a constant speed of 25 m s^-1, find the power of the car’s engine.
2. The engine’s power is now suddenly increased to 39 kW. Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.

(b) The car and caravan now travel up a straight hill, inclined at an angle of sin^-1 0.05 to the horizontal, at a constant speed of v m s^-1. The car’s engine is working at 32.5 kW.

Find v.