Past Exam Questions

An elevator moves vertically, supported by a cable. The diagram shows a velocity-time graph which models the motion of the elevator. The graph consists of 7 straight line segments.

The elevator accelerates upwards from rest to a speed of 2 m/s^-1 over a period of 1.5 s and then travels at this speed for 4.5 s, before decelerating to rest over a period of 1 s.

The elevator then remains at rest for 6 s, before accelerating to a speed of V m/s^-1 downwards over a period of 2 s. The elevator travels at this speed for a period of 5 s, before decelerating to rest over a period of 1.5 s.

(a) Find the acceleration of the elevator during the first 1.5 s.

(b) Given that the elevator starts and finishes its journey on the ground floor, find V.

(c) The combined weight of the elevator and passengers on its upward journey is 1500 kg. Assuming that there is no resistance to motion, find the tension in the elevator cable on its upward journey when the elevator is decelerating.

A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.

(i) Find the velocity of the particle at the instant when it reaches the surface of the water.

(ii) Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank.

(iii) Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water.

A particle of mass 3 kg falls from rest at a point 5 m above the surface of a liquid which is in a container. There is no instantaneous change in speed of the particle as it enters the liquid. The depth of the liquid in the container is 4 m. The downward acceleration of the particle while it is moving in the liquid is 5.5 m s^-2.

1. Find the resistance to motion of the particle while it is moving in the liquid.
2. Sketch the velocity-time graph for the motion of the particle, from the time it starts to move until the time it reaches the bottom of the container. Show on your sketch the velocity and the time when the particle enters the liquid, and when the particle reaches the bottom of the container.

An elevator is pulled vertically upwards by a cable. The velocity-time graph for the motion is shown above. Find

1. the distance travelled by the elevator,
2. the acceleration during the first stage and the deceleration during the third stage.

The mass of the elevator is 800 kg and there is a box of mass 100 kg on the floor of the elevator.

3. Find the tension in the cable in each of the three stages of the motion.
4. Find the greatest and least values of the magnitude of the force exerted on the box by the floor of the elevator.

The velocity-time graph shown models the motion of a parachutist falling vertically. There are four stages in the motion:

• falling freely with the parachute closed,
• decelerating at a constant rate with the parachute open,
• falling with constant speed with the parachute open,
• coming to rest instantaneously on hitting the ground.

(i) Show that the total distance fallen is 1048 m.

The weight of the parachutist is 850 N.

(ii) Find the upward force on the parachutist due to the parachute, during the second stage.

The diagram shows the velocity-time graph for the motion of a small stone which falls vertically from rest at a point A above the surface of liquid in a container. The downward velocity of the stone t s after leaving A is v m s^-1. The stone hits the surface of the liquid with velocity 7 m s^-1 when t = 0.7. It reaches the bottom of the container with velocity 5 m s^-1 when t = 1.2.

(i) Find

1. (a) the height of A above the surface of the liquid,
2. (b) the depth of liquid in the container.

(ii) Find the deceleration of the stone while it is moving in the liquid.

(iii) Given that the resistance to motion of the stone while it is moving in the liquid has magnitude 0.7 N, find the mass of the stone.

A ring of mass 4 kg is threaded on a smooth circular rigid wire with centre C. The wire is fixed in a vertical plane and the ring is kept at rest by a light string connected to A, the highest point of the circle. The string makes an angle of 25° to the vertical (see diagram).

Find the tension in the string and the magnitude of the normal reaction of the wire on the ring.

The diagram shows a block of mass 10 kg suspended below a horizontal ceiling by two strings AC and BC, of lengths 0.8 m and 0.6 m respectively, attached to fixed points on the ceiling. Angle ACB = 90°. There is a horizontal force of magnitude F N acting on the block. The block is in equilibrium.

\((a) In the case where F = 20, find the tensions in each of the strings.\)

(b) Find the greatest value of F for which the block remains in equilibrium in the position shown.

A small ring R is attached to one end of a light inextensible string of length 70 cm. A fixed rough vertical wire passes through the ring. The other end of the string is attached to a point A on the wire, vertically above R. A horizontal force of magnitude 5.6 N is applied to the point J of the string 30 cm from A and 40 cm from R. The system is in equilibrium with each of the parts AJ and JR of the string taut and angle AJR equal to 90° (see diagram).

1. Find the tension in the part AJ of the string, and find the tension in the part JR of the string. [5]
2. The ring R has mass 0.2 kg and is in limiting equilibrium, on the point of moving up the wire.
3. Show that the coefficient of friction between R and the wire is 0.341, correct to 3 significant figures. [4]
4. A particle of mass m kg is attached to R and R is now in limiting equilibrium, on the point of moving down the wire.
5. Given that the coefficient of friction is unchanged, find the value of m. [3]

A small ring of mass 0.2 kg is threaded on a fixed vertical rod. The end A of a light inextensible string is attached to the ring. The other end C of the string is attached to a fixed point of the rod above A. A horizontal force of magnitude 8 N is applied to the point B of the string, where AB = 1.5 m and BC = 2 m. The system is in equilibrium with the string taut and AB at right angles to BC (see diagram).

1. Find the tension in the part AB of the string and the tension in the part BC of the string.
2. The equilibrium is limiting with the ring on the point of sliding up the rod. Find the coefficient of friction between the ring and the rod.

A small smooth ring R, of mass 0.6 kg, is threaded on a light inextensible string of length 100 cm. One end of the string is attached to a fixed point A. A small bead B of mass 0.4 kg is attached to the other end of the string, and is threaded on a fixed rough horizontal rod which passes through A. The system is in equilibrium with B at a distance of 80 cm from A (see diagram).

1. Find the tension in the string.
2. Find the frictional and normal components of the contact force acting on B.
3. Given that the equilibrium is limiting, find the coefficient of friction between the bead and the rod.

A particle P of weight 5 N is attached to one end of each of two light inextensible strings of lengths 30 cm and 40 cm. The other end of the shorter string is attached to a fixed point A of a rough rod which is fixed horizontally. A small ring S of weight W N is attached to the other end of the longer string and is threaded on to the rod. The system is in equilibrium with the strings taut and AS = 50 cm (see diagram).

1. By resolving the forces acting on P in the direction of PS, or otherwise, find the tension in the longer string. [3]
2. Find the magnitude of the frictional force acting on S. [2]
3. Given that the coefficient of friction between S and the rod is 0.75, and that S is in limiting equilibrium, find the value of W. [3]

One end of a light inextensible string is attached to a fixed point A of a fixed vertical wire. The other end of the string is attached to a small ring B, of mass 0.2 kg, through which the wire passes. A horizontal force of magnitude 5 N is applied to the mid-point M of the string. The system is in equilibrium with the string taut, with B below A, and with angles ∠ABM and ∠BAM equal to 30° (see diagram).

(i) Show that the tension in BM is 5 N.

(ii) The ring is on the point of sliding up the wire. Find the coefficient of friction between the ring and the wire.

(iii) A particle of mass m kg is attached to the ring. The ring is now on the point of sliding down the wire. Given that the coefficient of friction between the ring and the wire is unchanged, find the value of m.

A ring of mass 0.2 kg is threaded on a fixed rough horizontal rod and a light inextensible string is attached to the ring at an angle \(\alpha\) above the horizontal, where \(\cos \alpha = 0.96\). The ring is in limiting equilibrium with the tension in the string \(T\) N (see diagram). Given that the coefficient of friction between the ring and the rod is 0.25, find the value of \(T\).

A ring of mass 4 kg is attached to one end of a light string. The ring is threaded on a fixed horizontal rod and the string is pulled at an angle of 25° below the horizontal (see diagram). With a tension in the string of \(T\) N the ring is in equilibrium.

(i) Find, in terms of \(T\), the horizontal and vertical components of the force exerted on the ring by the rod.

The coefficient of friction between the ring and the rod is 0.4.

(ii) Given that the equilibrium is limiting, find the value of \(T\).

A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at 45° to the horizontal (see diagram).

(i) Show that the normal component of the contact force acting on the ring has magnitude 3.05 N, correct to 3 significant figures.

(ii) The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod.

A ring of mass 1.1 kg is threaded on a fixed rough horizontal rod. A light string is attached to the ring and the string is pulled with a force of magnitude 13 N at an angle \(\alpha\) below the horizontal, where \(\tan \alpha = \frac{5}{12}\) (see diagram). The ring is in equilibrium.

(i) Find the frictional component of the contact force on the ring.

(ii) Find the normal component of the contact force on the ring.

(iii) Given that the equilibrium of the ring is limiting, find the coefficient of friction between the ring and the rod.

A ring of mass 0.3 kg is threaded on a horizontal rough rod. The coefficient of friction between the ring and the rod is 0.8. A force of magnitude 8 N acts on the ring. This force acts at an angle of 10° above the horizontal in the vertical plane containing the rod.

Find the time taken for the ring to move, from rest, 0.6 m along the rod.

The diagram shows a ring of mass 0.1 kg threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is 0.8. A force of magnitude \(T \text{ N}\) acts on the ring in a direction at \(30^\circ\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest.

(a) Find the greatest value of \(T\) for which the ring remains at rest. [4]

(b) Find the acceleration of the ring when \(T = 3\). [3]

A small ring of mass 0.024 kg is threaded on a fixed rough horizontal rod. A light inextensible string is attached to the ring and the string is pulled with a force of magnitude 0.195 N at an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac{5}{13}\). When the angle \(\theta\) is below the horizontal (see Fig. 1) the ring is in limiting equilibrium.

(i) Find the coefficient of friction between the ring and the rod.

When the angle \(\theta\) is above the horizontal (see Fig. 2) the ring moves.

(ii) Find the acceleration of the ring.