OCR A Level Further Maths B (MEI) Mechanics Minor (minor option - choose one with a major, or three alone) June 2023 question paper Y431

Show that the student’s formula is dimensionally consistent.

A car of mass 1400 kg, travels along a straight horizontal road AB, after which it descends a hill BC inclined at a constant angle of 7° to the horizontal (see diagram). A, B and C all lie in the same vertical plane. Throughout the entire journey, the total resistance to the car’s motion is constant.

Between A and B, the car moves at a constant speed of 12 m s^-1, and the power developed by the car is a constant P W. When the car reaches B, the engine is switched off and the car travels down a line of greatest slope from B to C with an acceleration of 0.8 m s^-2. The resistance to motion is unchanged.

Determine the value of P.

When the car reaches C it turns round and travels back up the hill towards B at a constant speed of v m s^-1. The power developed by the car between C and B is a constant 16 kW. The resistance to motion is unchanged.

Determine the value of v.

Determine the coefficient of restitution between P and Q.

Calculate the impulse exerted on P by Q during their collision.

After colliding with P, Q strikes the wall, which is perpendicular to the direction of the motion of Q, and comes to rest exactly halfway between A and B. The collision between Q and the wall is perfectly elastic.

Determine the coefficient of friction between Q and AB.

The diagram shows two particles P and Q, of masses 10 kg and 5 kg respectively, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. The pulley is fixed at the highest point A on a smooth curved surface, the vertical cross-section of which is a quadrant of a circle with centre O and radius 2 m. Particle Q hangs vertically below the pulley and P is in contact with the surface, where the angle AOP is equal to θ °. The pulley, P and Q all lie in the same vertical plane.

Throughout this question you may assume that there are no resistances to the motion of either P or Q and the force acting on P due to the tension in the string is tangential to the curved surface at P.

Given that P is in equilibrium at the point where θ = α, determine the value of α.

Particle P is now released from rest at the point on the surface where θ = 35, and starts to move downwards on the surface. In the subsequent motion it is given that P does not leave the surface.

By considering energy, determine the speed of P at the instant when θ = 45.

State one modelling assumption you have made in determining the answer to part (b).

Fig. 5.1 shows a particle P, of mass 5 kg, and a particle Q, of mass 11 kg, which are attached to the ends of a light, inextensible string. The string is taut and passes over a small smooth pulley fixed to the ceiling.

When a force of magnitude H N, acting at an angle θ to the upward vertical, is applied to Q the particles hang in equilibrium, with the part of the string connecting the pulley to Q making an angle of 40° with the upward vertical. It is given that the force acts in the same vertical plane in which the string lies.

Determine the values of H and θ.

Particle Q is now removed. The string is instead attached to one end of a uniform beam B of length 3 m and mass 7 kg. The other end of B is in contact with a rough horizontal floor. The situation is shown in Fig. 5.2.

With B in equilibrium, at an angle φ to the horizontal, the part of the string connecting the pulley to B makes an angle of 30° with the upward vertical.

It is given that the string and B lie in the same vertical plane.

Determine the smallest possible value for the coefficient of friction between B and the floor.

Determine the value of φ.

In this question you may use the fact that the volume of a sphere of radius r is r³.

Fig. 6.1 shows a container in the shape of an open-topped cylinder. The cylinder has height 18 cm and radius 4 cm. The curved surface and the base can be modelled as uniform laminae with the same mass per unit area. The container rests on a horizontal surface.

Show that the centre of mass of the container lies 8.1 cm above its base.

The mass of the container is 400 grams. Water is poured into the container to reach a height of h cm above the base. The centre of mass of the combined container and water lies y cm above the base. Water has a density of 1 gram per cm³.

In this question you must show detailed reasoning.

By formulating an expression for y in terms of h, determine the value of h for which y is lowest.

More water is now poured into the container. A sphere of radius 3 cm is placed into the container, where it sinks to the bottom. The surface of the water is now 4.5 cm from the top of the container, as shown in Fig. 6.2.

Show that the centre of mass of the water in the container lies 7.5 cm above the base of the

container.

The sphere has a density of 4 grams per cm³.

The centre of mass of the combined container, water and sphere lies z cm above the base.

Determine the value of z.