OCR GCE AS Level Further Maths B (MEI) Mechanics a (optional - choose any two of Y411/Y412/Y413/Y414) June 2025 question paper Y411

A system consists of an object performing oscillations whilst suspended at the end of a spring S.

The period, t, is defined as the time taken for the object to complete one oscillation. The period is given by the formula

where

• λ is a constant,

• m is the mass of the object,

• k is the stiffness of S.

One possible unit which k can be measured in is kg s^–2.

Show that λ is dimensionless.

For a given object P and S, the period of one oscillation is measured to be 1.48 seconds, correct to 3 significant figures. The object P is replaced with another object Q which has three times the mass of P.

Find an estimate for the period of one oscillation of Q and S.

Show that F = 35.

Draw a triangle of forces to represent the system shown in the diagram.

Determine the value of θ. Give your answer correct to 3 significant figures.

The value of θ is now increased slightly.

Explain what happens to the block.

In parts (a) and (b) of this question you should assume that the car experiences no resistances to motion.

A car of mass 1200 kg is moving on a straight road.

At first the car travels up a section of the road inclined at 3° to the horizontal.

Determine the power developed by the car at the instant when it has a speed of 22 m s^-1 and

is accelerating at 0.4 m s^-2.

The maximum power that can be developed by the car is 40 kW.

Calculate the greatest speed that can be maintained by the car when travelling up the road.

State one way in which the modelling of the car’s motion in this question could be refined.

Two small trucks, A and B, are at rest on a smooth horizontal track. The mass of A is 2 kg, and the mass of B is 3 kg. At one end of the track there is a fixed bumper, as shown in the diagram. When the trucks touch, they become permanently attached to each other and form a single object AB.

Truck A is set in motion towards the bumper with speed 8 m s^-1. At the same instant, truck B is also set in motion towards the bumper with speed 4 m s^-1.

State the initial impulse required to set A in motion.

You are given the following information.

• The coefficient of restitution between B and the bumper is e.

• If A collides with B before B reaches the bumper, the speed of AB after it hits the bumper is v m s^-1.

• If B collides with the bumper first and then collides with A, after this second collision AB moves towards the bumper with speed v m s^-1.

• All impacts are direct.

Determine the following in either order.

• The value of e

• The value of v

Fig 5.1 shows a uniform triangular lamina OAB where O is the origin of the coordinate system in which the points A and B have coordinates (30, 0) and (90, 50) respectively.

Calculate the coordinates of the centre of mass of OAB.

The lamina OAB is the cross-section through the centre of mass of a uniform solid prism S.

Explain why S can not rest in equilibrium if OA is placed on a horizontal surface.

The prism S now rests in equilibrium on a rough plane inclined at an angle of θ ° to the horizontal, as shown in Fig. 5.2. The line OA lies along a line of greatest slope of the plane.

Assuming that S does not slide, calculate the least possible value of θ.

Prism S is now placed on rough horizontal ground so that OA is horizontal, but with B in contact with a smooth vertical wall, as shown in Fig. 5.3. The vertical plane containing O, A and B is perpendicular to the wall.

The coefficient of friction between S and the ground is μ. You are given that S is in limiting equilibrium with the total contact force exerted by the ground on S acting at A.

Determine the value of μ.

Throughout this question you should assume that there is no air resistance.

A particle is projected vertically upwards with speed u m s^-1.

Show that the particle will reach a height of 3 m above the point of projection if

u² ⩾ 58.8.

The diagram shows a horizontal floor and a horizontal ceiling. The vertical distance between the floor and ceiling is 3 m. A particle is held 2.5 m above the floor and is projected vertically downwards with speed 8 m s^-1. The coefficient of restitution between the particle and the floor is 0.95. The coefficient of restitution between the particle and the ceiling is 1.

Show that the particle will have a speed of 10.1 m s^-1 immediately after the first impact with

the floor, correct to 3 significant figures.

Hence, using part (a), show that the particle will reach the ceiling after the first impact with

the floor.

Explain why the speed of the particle immediately before the second impact with the floor is

the same as the speed of the particle immediately after the first impact with the floor.

In this question you must show detailed reasoning.

Find the number of times the particle hits the ceiling.

Determine the values of p, q and r.

The three forces are removed. The lamina is in the same position as shown in the diagram.

Three particles are now attached to certain points on the lamina as follows.

• A particle of mass 2 kg at the point (λ, λ²), where 0 ⩽ λ ⩽ 1

• A particle of mass 3 kg at the point A

• A particle of mass 1 kg at the point C

Find the coordinates of the composite body’s centre of mass. Give your answer in terms

of λ.

Show that, as λ varies, the centre of mass of the combined object lies on the curve

y = 3x² - 3x + k,

where k is a constant to be determined.