OCR A Level Further Maths B (MEI) Mechanics Major (major option - choose one of Y421/Y422, or three minors) June 2023 question paper Y421

A car of mass 800 kg moves in a straight line along a horizontal road.

There is a constant resistance to the motion of the car of magnitude 600 N.

When the car is travelling at a speed of 15 m s^-1 the power developed by the car is 27 kW.

Determine the acceleration of the car when it is travelling at 15 m s^-1.

Two small uniform smooth spheres A and B have masses 0.5 kg and 2 kg respectively. The two spheres are travelling in the same direction in the same straight line on a smooth horizontal surface. Sphere A is moving towards B with speed 6 m s^-1 and B is moving away from A with speed 2 m s^-1 (see diagram). Spheres A and B collide. After this collision A moves with speed 0.2 m s^-1.

Determine the possible speeds with which B moves after the collision.

Find the greatest possible value of θ, giving your answer to the nearest degree.

A rigid lamina of negligible mass is in the form of a rhombus ABCD, where AC = 6 m and BD = 8 m. Forces of magnitude 2 N, 4 N, 3 N and 5 N act along its sides AB, BC, CD and DA, respectively, as shown in the diagram. A further force F N, acting at A, and a couple of magnitude G Nm are also applied to the lamina so that it is in equilibrium.

Determine the magnitude and direction of F.

Determine the value of G.

A particle P of mass m kg is projected with speed u m s^-1 along a rough horizontal surface. During the motion of P, a constant frictional force of magnitude F N acts on P. When the velocity of P is v m s^-1, it experiences a force of magnitude kv N due to air resistance, where k is a constant.

Determine the dimensions of k.

At time t seconds, where t ⩾ 0, a particle P has position vector r metres, where

r = (2t² - 12t + 6)i + (t³ + 3t² - 8t)j.

The velocity of P at time t seconds is v m s^-1.

Find v in terms of t.

Determine the speed of P at the instant when it is moving parallel to the vector i - 4 j.

Determine the value of t when the magnitude of the acceleration of P is 20.2 m s^-2.

One end of a rope is attached to a block A of mass 2 kg. The other end of the rope is attached to a second block B of mass 4 kg. Block A is held at rest on a fixed rough ramp inclined at 30° to the horizontal. The rope is taut and passes over a small smooth pulley P which is fixed at the top of the ramp. The part of the rope from A to P is parallel to a line of greatest slope of the ramp. Block B hangs vertically below P, at a distance d m above the ground, as shown in the diagram.

Block A is more than d m from P. The blocks are released from rest and A moves up the ramp.

The blocks are modelled as particles, the rope is modelled as light and inextensible, and air resistance can be ignored.

Determine, in terms of g and d, the work done against friction as A moves d m up the ramp.

Given that the speed of B immediately before it hits the ground is 1.75 m s^-1, use the

work–energy principle to determine the value of d.

Suggest one improvement, apart from including air resistance, that could be made to the

model to make it more realistic.

Region R is occupied by a uniform lamina.

Determine, in terms of k, an expression for the y-coordinate of the centre of mass of the lamina. Give your answer in the form , where a, b, c and d are integers to be

determined.

In this question take g = 10.

A small ball P is projected with speed 20 m s^-1 at an angle of elevation of (α + θ) from a point O at the bottom of a smooth plane inclined at an angle α to the horizontal, where tan α = and tan θ = . The ball subsequently hits the plane at a point A, where OA is a line of greatest slope of the plane, as shown in the diagram.

Determine the following, in either order.

• The components of the velocity of P, parallel and perpendicular to the plane, immediately before P hits the plane at A.

• The distance OA.

After P hits the plane at A it continues to move away from O. Immediately after hitting the plane at A the direction of motion of P makes an angle β with the horizontal.

Determine the maximum possible value of β, giving your answer to the nearest degree.

A hollow sphere has centre O and internal radius r. A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O.

The point A lies on the rim of the bowl such that AO makes an angle of 30° with the horizontal (see diagram).

The motion of P takes place in the vertical plane containing O and A. The particle P passes through a point B on the inner surface, where OB makes an acute angle θ with the vertical.

Determine, in terms of m, g, u, r and θ, the magnitude of the force exerted on P by the bowl

when P is at B.

The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is 4 mg.

Determine, in terms of r, the vertical distance of B above the floor.

It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.

Show that u² > 2gr.

Show that the least value of P needed to move the prism is given by

Determine the range in which the value of θ must lie.

Two small uniform smooth spheres A and B are of equal radius and have masses m and λ m respectively. The spheres are on a smooth horizontal surface.

Sphere A is moving on the surface with velocity u₁i + u₂j towards B, which is at rest. The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to i.

The coefficient of restitution between A and B is e.

Explain why, when the spheres collide, the impulse of A on B is in the direction of i.

Determine this impulse in terms of λ, m, e and u₁.

The loss in kinetic energy due to the collision between A and B is mu₁².

Determine the range of possible values of λ.

A particle P of mass m is fixed to one end of a light spring of natural length a and modulus of elasticity man², where n > 0. The other end of the spring is attached to the ceiling of a lift. The lift is at rest and P is hanging vertically in equilibrium.

Find, in terms of g and n, the extension in the spring.

At time t = 0 the lift begins to accelerate upwards from rest. At time t, the upward displacement of the lift from its initial position is y and the extension of the spring is x.

Express, in terms of g, n, x and y, the upward displacement of P from its initial position at

time t.

Verify that x = knt + gn - k sin(nt)).