Past Exam Questions

(i) A particle P of mass 1.2 kg is released from rest at the top of a slope and starts to move. The slope has length 4 m and is inclined at 25° to the horizontal. The coefficient of friction between P and the slope is \(\frac{1}{4}\). Find

1. the frictional component of the contact force on P,
2. the acceleration of P,
3. the speed with which P reaches the bottom of the slope.

(ii) After reaching the bottom of the slope, P moves freely under gravity and subsequently hits a horizontal floor which is 3 m below the bottom of the slope.

1. Find the loss in gravitational potential energy of P during its motion from the bottom of the slope until it hits the floor.
2. Find the speed with which P hits the floor.

A block of mass 10 kg is at rest on a rough plane inclined at an angle of 30° to the horizontal. A force of 120 N is applied to the block at an angle of 20° above a line of greatest slope (see diagram). There is a force resisting the motion of the block and 200 J of work is done against this force when the block has moved a distance of 5 m up the plane from rest.

Find the speed of the block when it has moved a distance of 5 m up the plane from rest.

A man has mass 80 kg. He runs along a horizontal road against a constant resistance force of magnitude \(P\) N. The total work done by the man in increasing his speed from 4 m s\(^{-1}\) to 5.5 m s\(^{-1}\) while running a distance of 60 metres is 1200 J. Find the value of \(P\).

A man pushes a wheelbarrow of mass 25 kg along a horizontal road with a constant force of magnitude 35 N at an angle of 20° below the horizontal. There is a constant resistance to motion of 15 N. The wheelbarrow moves a distance of 12 m from rest.

(i) Find the work done by the man.

(ii) Find the speed attained by the wheelbarrow after 12 m.

One end of a light inextensible string is attached to a block. The string makes an angle of \(\theta^\circ\) with the horizontal. The tension in the string is 20 N. The string pulls the block along a horizontal surface at a constant speed of 1.5 m s\(^{-1}\) for 12 s. The work done by the tension in the string is 50 J. Find \(\theta\).

A particle of mass 0.6 kg is dropped from a height of 8 m above the ground. The speed of the particle at the instant before hitting the ground is 10 m s^-1. Find the work done against air resistance.

A girl on a sledge starts, with a speed of 5 m/s, at the top of a slope of length 100 m which is at an angle of 20° to the horizontal. The sledge slides directly down the slope.

1. Given that there is no resistance to the sledge’s motion, find the speed of the sledge at the bottom of the slope.
2. It is given instead that the sledge experiences a resistance to motion such that the total work done against the resistance is 8500 J, and the speed of the sledge at the bottom of the slope is 21 m/s. Find the total mass of the girl and the sledge.

A cyclist has mass 85 kg and rides a bicycle of mass 20 kg. The cyclist rides along a horizontal road against a total resistance force of 40 N. Find the total work done by the cyclist in increasing his speed from 5 m s^-1 to 10 m s^-1 while travelling a distance of 50 m.

One end of a light inextensible string is attached to a block. The string makes an angle of 60° above the horizontal and is used to pull the block in a straight line on a horizontal floor with acceleration 0.5 m/s². The tension in the string is 8 N. The block starts to move with speed 0.3 m/s. For the first 5 s of the block’s motion, find

1. the distance travelled,
2. the work done by the tension in the string.

A block B of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(\theta\) above the horizontal. The normal component of the contact force acting on B has magnitude 20 N.

(i) Show that \(\sin \theta = 0.28\).

(ii) Find the work done by the pulling force in moving the block a distance of 5 m.

A small block of mass 3 kg is initially at rest at the bottom O of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). A force of magnitude 35 N acts on the block at an angle \(\beta\) above the plane, where \(\sin \beta = 0.28\) and \(\cos \beta = 0.96\). The block starts to move up a line of greatest slope of the plane and passes through a point A with speed 4 m s\(^{-1}\). The distance \(OA\) is 12.5 m (see diagram).

(i) For the motion of the block from \(O\) to \(A\), find the work done against the frictional force acting on the block.

(ii) Find the coefficient of friction between the block and the plane.

At the instant that the block passes through \(A\) the force of magnitude 35 N ceases to act.

(iii) Find the distance the block travels up the plane after passing through \(A\).

Particles A of mass 0.4 kg and B of mass 1.6 kg are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. A is held at rest and B hangs freely, with both straight parts of the string vertical and both particles at a height of 1.2 m above the floor (see diagram). A is released and both particles start to move.

1. Find the work done on B by the tension in the string, as B moves to the floor.
2. When particle B reaches the floor it remains at rest. Particle A continues to move upwards. Find the greatest height above the floor reached by particle A.

A block of mass 15 kg slides down a line of greatest slope of an inclined plane. The top of the plane is at a vertical height of 1.6 m above the level of the bottom of the plane. The speed of the block at the top of the plane is 2 m/s^-1 and the speed of the block at the bottom of the plane is 4 m/s^-1.

Find the work done against the resistance to motion of the block.

A box of mass 25 kg is pulled in a straight line along a horizontal floor. The box starts from rest at a point A and has a speed of 3 m/s when it reaches a point B. The distance AB is 15 m. The pulling force has magnitude 220 N and acts at an angle of \(\alpha^\circ\) above the horizontal. The work done against the resistance to motion acting on the box, as the box moves from A to B, is 3000 J. Find the value of \(\alpha\).

A lorry of mass 15,000 kg climbs from the bottom to the top of a straight hill, of length 1440 m, at a constant speed of 15 m s^-1. The top of the hill is 16 m above the level of the bottom of the hill. The resistance to motion is constant and equal to 1800 N.

(i) Find the work done by the driving force.

On reaching the top of the hill the lorry continues on a straight horizontal road and passes through a point P with speed 24 m s^-1. The resistance to motion is constant and is now equal to 1600 N. The work done by the lorry’s engine from the top of the hill to the point P is 5030 kJ.

(ii) Find the distance from the top of the hill to the point P.

A block B lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N, making angles of \(\alpha\) and \(\beta\) respectively with the x-direction, act on B as shown in the diagram, and B is moving in the x-direction with constant speed. It is given that \(\cos \alpha = 0.6\) and \(\cos \beta = 0.8\).

(i) Find the total work done by the forces shown in the diagram when B has moved a distance of 20 m.

(ii) Given that the coefficient of friction between the block and the plane is \(\frac{5}{8}\), find the weight of the block.

A and B are two points 50 metres apart on a straight path inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.05\), with A above the level of B. A block of mass 16 kg is pulled down the path from A to B. The block starts from rest at A and reaches B with a speed of 10 m s^-1. The work done by the pulling force acting on the block is 1150 J.

1. Find the work done against the resistance to motion.

The block is now pulled up the path from B to A. The work done by the pulling force and the work done against the resistance to motion are the same as in the case of the downward motion.

2. Show that the speed of the block when it reaches A is the same as its speed when it started at B.

A car of mass 1250 kg travels from the bottom to the top of a straight hill of length 600 m, which is inclined at an angle of 2.5° to the horizontal. The resistance to motion of the car is constant and equal to 400 N. The work done by the driving force is 450 kJ. The speed of the car at the bottom of the hill is 30 m s^-1. Find the speed of the car at the top of the hill.

A block is pushed along a horizontal floor by a force of magnitude 45 N acting at an angle of 14° to the horizontal (see diagram). Find the work done by the force in moving the block a distance of 25 m.

A ring is threaded on a fixed horizontal bar. The ring is attached to one end of a light inextensible string which is used to pull the ring along the bar at a constant speed of 0.5 m s^-1. The string makes a constant angle of 24° with the bar and the tension in the string is 6 N (see diagram). Find the work done by the tension in a period of 8 s.