OCR A Level Further Maths B (MEI) Mechanics Minor (minor option - choose one with a major, or three alone) June 2024 question paper Y431

A car of mass 1500 kg travels along a horizontal straight road. There are no resistances to the car’s motion. The power developed by the car as it increases its speed from 20 m s^-1 to 30 m s^-1 over t seconds is a constant 5000 W.

Determine the value of t.

Find the acceleration of the car when its speed is 25 m s^-1.

State the dimensions of force.

Use the following metric-imperial conversion factors for the rest of this question.

• 1 kg = 2.2 lb (pounds)

• 1 m = 39.4 in (inches)

A unit of force used in the imperial system is the pound-force (lbf). 1 lbf is defined as the gravitational force exerted on 1 lb on the surface of the Earth.

Show that 1 lbf is approximately equal to 4.45 N.

The pascal (Pa) is a unit of pressure equivalent to 1 Newton per square metre. Pressure can also be measured in pound-force per square inch (psi).

A diver, at a depth of 40 m, experiences a typical pressure of 5 × 10⁵ Pa.

Determine whether this is greater or less than the pressure in a bicycle tyre of 80 psi.

In various physical contexts, energy density is the amount of energy stored in a given region of space per unit volume.

Show that energy density and pressure are dimensionally equivalent.

The diagram shows the three points A, B and C that lie along a line of greatest slope on a rough plane which is inclined at an angle of 25 to the horizontal.

A block of mass 6 kg is placed at B and is projected up the plane towards C with an initial speed of u m s^-1. The block travels 3.5 m before coming instantaneously to rest at C, before sliding back down the plane. When the block is sliding back down the plane it attains its initial speed at A, which lies x m down the plane from B.

It is given that the work done against resistance throughout the motion is 4 joules per metre.

Use an energy method to determine the following.

The value of u

The value of x

A student claims that half of the energy lost due to resistances is accounted for by friction between the block and the plane, and the other half by air resistance.

Assuming that the student’s claim is correct, determine the coefficient of friction between the

block and the plane.

Fig. 4.1 shows two spheres, A and B, on a smooth horizontal surface. Their masses are 3 kg and 1 kg respectively.

Initially, sphere A travels at a speed of 1 m s^-1 in a straight line towards B, which is at rest. The spheres collide and the coefficient of restitution between A and B is e.

During the collision, the kinetic energy of the system decreases by 21%.

Determine the value of e.

State why in part (a) it was necessary to assume that A and B have equal radii.

A uniform lamina OABC is in the shape of a trapezium where O is the origin of the coordinate system in which the points A, B and C have coordinates (120, 0), (60, 90) and (30, 90) respectively (see diagram). The units of the axes are centimetres.

The centre of mass of the lamina lies at ( .

Show that = 54 and determine the value of

The lamina is placed horizontally so that it rests on three supports, whose points of contact are at B, C and D, where D lies on the edge OA and has coordinates (d, 0).

Determine the range of values of d for the lamina to rest in equilibrium.

It is now given that d = 63, and that the lamina has a weight of 100 N.

Determine the forces exerted on the lamina by each of the supports at B, C and D.

Fig. 6.1 shows three forces of magnitude 15 N, 15 N and 30 N acting on a rigid beam AB of length 6 m. One of the forces of magnitude 15 N acts at A, and the other force of magnitude 15 N acts at B. The force of magnitude 30 N acts at distance of x m from B. All three forces act in a direction perpendicular to the beam as shown in Fig. 6.1. The beam and the three forces all lie in the same horizontal plane. The three forces form a couple of magnitude 42 N m in the clockwise direction.

Determine the value of x.

Fig. 6.2 shows the same beam, without the three forces from Fig. 6.1, resting in limiting equilibrium against a step. The point of contact, C, between the beam and the edge of the step lies 1.5 m from A. The other end of the beam rests on a horizontal floor. The contacts between the beam and both the step and the floor are rough.

Draw a diagram to show all the forces acting on the beam.

The coefficient of friction between the beam and the step and the coefficient of friction between the beam and the floor are the same, and are denoted by μ.

Show that μ² - 6μ + 1 = 0.

Hence determine the value of μ.