OCR GCE AS Level Further Maths B (MEI) Mechanics a (optional - choose any two of Y411/Y412/Y413/Y414) June 2023 question paper Y411

Throughout all parts of this question, the resistance to the motion of a car has magnitude kv² N, where v m s^-1 is the speed of the car and k is a constant.

At first, the car travels along a straight horizontal road with constant speed 20 m s^-1. The power developed by the car at this speed is 5000 W.

Find the power the car must develop in order to maintain a constant speed of 28 m s^-1 when

travelling along the same horizontal road.

The car climbs a hill which is inclined at an angle of 2° to the horizontal. The power developed by the car is 13 000 W, and the car has a constant speed of 20 m s^-1.

Determine the mass of the car.

A ball P of mass m kg is held at a height of 12.8 m above a horizontal floor. P is released from rest and rebounds from the floor. After the first bounce, P reaches a maximum height of 5 m above the floor.

Two models, A and B, are suggested for the motion of P.

Model A assumes that air resistance may be neglected.

Determine, according to model A, the coefficient of restitution between P and the floor.

Model B assumes that the collision between P and the floor is perfectly elastic, but that work is done against air resistance at a constant rate of E joules per metre.

Show that, according to model B, E = mg.

Show that both models predict that P will attain the same maximum height after the second

bounce.

The time period T of a satellite in circular orbit around a planet satisfies the equation

GMT² = 4π²R³,

where

• G is the universal gravitational constant,

• M is the mass of the planet,

• R is the radius of the orbital circle.

Calculate, according to the student’s model, how many times greater the approach speed is between a pair of stars which are 6.13 light-years apart and the same pair of stars if they were

8.64 light-years apart. (A light-year is a unit of distance.)

The diagram shows three beads, A, B and C, of masses 0.3 kg, 0.5 kg and 0.7 kg respectively, threaded onto a smooth wire circuit consisting of two straight and two semi-circular sections. The circuit occupies a vertical plane, with the two straight sections horizontal and the upper section 0.45 m directly above the lower section.

Initially, the beads are at rest. A and B are each given an impulse so that they move towards each other, A with a speed of 8 m s^-1 and B with a speed of 1.6 m s^-1. In the subsequent collision between A and B, A is brought to rest.

Show that the speed of B before this collision is 4.37 m s^-1, correct to 3 significant figures.

In this collision between B and C, B is brought to rest.

Determine whether C next collides with A or with B.

Explain why, if B has a greater mass than C, B could not be brought to rest in their collision.

Fig. 5.1 shows the uniform cross-section of a solid S which is formed from a cylinder by boring two cylindrical tunnels the entire way through the cylinder. The radius of S is 50 cm, and the two tunnels have radii 10 cm and 30 cm.

The material making up S has uniform density.

Coordinates refer to the axes shown in Fig. 5.1 and the units are centimetres.

Show that = 12 and find the value of

Solid S is placed onto two rails, A and B, whose point of contacts with S are at (- 30, - 40) and (30, - 40) as shown in Fig. 5.2. Two points, P (0, 50) and Q (0, - 50), are marked on Fig. 5.2.

At first, you should assume that the contact between S and the two rails is smooth.

Determine the angle PQ makes with the vertical, after S settles into equilibrium.

For the remainder of the question, you should assume that the contact between S and A is rough, that the contact between S and B is smooth, and that S does not move when placed on the rails. Fig. 5.3 shows only the forces exerted on S by the rails. The normal contact forces exerted by A and B on S have magnitude R_A N and R_B N respectively. The frictional force exerted by A on S has magnitude F N.

The weight of S is W N.

By taking moments about the origin, express F in the form λ W, where λ is a constant to be

determined.

Given that S is in limiting equilibrium, find the coefficient of friction between A and S.

A uniform beam of length 6 m and mass 10 kg rests horizontally on two supports A and B, which are 3.8 m apart. A particle P of mass 4 kg is attached 1.95 m from one end of the beam (see Fig. 6.1).

Fig. 6.1

When A is x m from the end of the beam, the supports exert forces of equal magnitude on the beam.

Determine the value of x.

P is now removed. The same beam is placed on the supports so that B is 0.7 m from the end of the beam. The supports remain 3.8 m apart (see Fig. 6.2).

Fig. 6.2

The contact between A and the beam is smooth. The contact between B and the beam is rough,

with coefficient of friction 0.4.

A small force of magnitude T N is applied to one end of the beam. The force acts in the same vertical plane as the beam and the angle the force makes with the beam is 60°.

As T is increased, forces T_L and T_S are defined in the following way.

• T_L is the value of T at which the beam would start lifting, assuming that is not already sliding.

• T_S is the value of T at which the beam would start sliding, assuming that it has not already lifted.

Show that T_L = 49.1, correct to 3 significant figures.

Determine whether the beam will first slide or lift.