OCR A Level Further Maths B (MEI) Mechanics Major (major option - choose one of Y421/Y422, or three minors) November 2020 question paper Y421

A particle P of mass 0.5 kg is attached to a fixed point O by a light elastic string of natural length 3 m and modulus of elasticity 75 N.

P is released from rest at O and is allowed to fall freely.

Determine the length of the string when P is at its lowest point in the subsequent motion.

The vertices of a triangular lamina, which is in the x – y plane, are at the origin O and the points A (2, 3) and B (- 2, 1).

Forces 2 i + j and - 3 i + 2 j are applied to the lamina at A and B, respectively, and a force F, whose line of action is in the x – y plane, is applied at O.

The three forces form a couple.

Determine the magnitude and the direction of F.

Determine the magnitude and direction of the additional couple that must be applied to the

lamina in order to keep it in equilibrium.

A particle P moves so that its position vector r at time t is given by

r = (5 + 20t)i + (95 + 10t - 5t²)j.

Determine the initial velocity of P.

At time t = T, P is moving in a direction perpendicular to its initial direction of motion.

Determine the value of T.

Determine the distance of P from its initial position at time T.

A car of mass 900 kg moves along a straight level road. The power developed by the car is constant and equal to 60 kW. The resistance to the motion of the car is constant and equal to 1500 N.

At time t seconds the velocity of the car is denoted by v m s^–1. Initially the car is at rest.

A small ball of mass m kg is held at a height of 78.4 m above horizontal ground. The ball is released from rest, falls vertically and rebounds from the ground. The coefficient of restitution between the ball and ground is e.

The ball continues to bounce until it comes to rest after 6 seconds.

Determine the value of e.

Given that the magnitude of the impulse that the ground exerts on the ball at the first bounce is

23.52 Ns, determine the value of m.

A particle P of mass m is attached to one end of a light elastic string of natural length 6 a and modulus of elasticity 3 mg. The other end of the string is fixed to a point O on a smooth plane, which is inclined at an angle of 30° to the horizontal. The string lies along a line of greatest slope of the plane and P rests in equilibrium on the inclined plane at a point A, as shown in Fig. 7.

P is now pulled a further distance 2 a down the line of greatest slope through A and released from rest. At time t later, the displacement of P from A is x, where the positive direction of x is down the plane.

Show that, until the string slackens, x satisfies the differential equation

Determine, in terms of a and g, the time at which the string slackens.

Find, in terms of a and g, the speed of P when the string slackens.

8 [In this question, you may use the fact that the volume of a right circular cone of base radius r and

By using integration, show that the centre of mass of a uniform solid right circular cone of

height h and base radius r is at a distance from the vertex.

Fig. 8 shows the side view of a toy formed by joining a uniform solid circular cylinder of radius r and height 2 r to a uniform solid right circular cone, made of the same material as the cylinder, of radius r and height r.

The toy is placed on a horizontal floor with the curved surface of the cone in contact with the floor.

Determine whether the toy will topple.

Explain why it is not necessary to know whether the floor is rough or smooth in answering

part (b).

Fig. 9 shows a uniform rod AB of length 2 a and weight 8 W which is smoothly hinged at the end A to a point on a fixed horizontal rough bar. A small ring of weight W is threaded on the bar and is connected to the rod at B by a light inextensible string of length 2 a. The system is in equilibrium with the rod inclined at an angle θ to the horizontal.

Determine, in terms of W and θ, the tension in the string.

It is given that, for equilibrium to be possible, the greatest distance the ring can be from A is 2.4 a

Determine the coefficient of friction between the bar and the ring.

Fig. 10 shows a small bead P of mass m which is threaded on a smooth thin wire. The wire is in the form of a circle of radius a and centre O. The wire is fixed in a vertical plane.

The bead is initially at the lowest point A of the wire and is projected along the wire with a velocity which is just sufficient to carry it to the highest point on the wire. The angle between OP and the downward vertical is denoted by θ.

Hence determine, in terms of g and a, the angular acceleration of P when θ takes the value

found in part (a).

Two uniform small smooth spheres A and B have equal radii and equal masses. The spheres are on a smooth horizontal surface. Sphere A is moving at an acute angle α to the line of centres, when it collides with B, which is stationary.

After the impact A is moving at an acute angle β to the line of centres. The coefficient of restitution

Show that tan β = 3 tan α.