Edexcel A Level Further Maths M3 · Mechanics 3 June 2013 question paper 6679R

A hollow right circular cone, of base radius a and height h, is fixed with its axis vertical and vertex downwards, as shown in Figure 1. A particle moves with constant speed v in a horizontal circle of radius a on the smooth inner surface of the cone.

A particle of mass 4 kg is moving along the horizontal x-axis under the action of a single force which acts in the positive x-direction. At time t seconds the force has magnitude

When t = 0 the particle has speed 2 m s^–1 in the positive x-direction. Find the work done by the force in the interval 0 ≤ t 4

A particle P of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2 m and modulus of elasticity 20 N. The other end of the spring is attached to a fixed point A. The particle P is held at rest at the point B, which is 1 m vertically below A, and then released.

Find the acceleration of P immediately after it is released from rest.

The particle comes to instantaneous rest for the first time at the point C.

Find the distance BC.

A particle P is moving along the positive x-axis. At time t seconds, t 0, P is x metres from the origin O and is moving away from O with velocity v m s^–1, where v = . When t = 0, P is at O. Find

the distance of P from O when t = 2

the magnitude and direction of the acceleration of P when t = 2

Part of a hollow spherical shell, centre O and radius r, forms a bowl with a plane circular rim. The bowl is fixed to a horizontal surface at A with the rim uppermost and horizontal. The point A is the lowest point of the bowl. The point B, where AOB = and tan is on the rim of the bowl, as shown in Figure 2. A small smooth marble M is placed inside the bowl at Agr). The motion of M takes place in the vertical plane OAB.

Show that the speed of M as it reaches B is gr. After leaving the surface of the bowl at B, M moves freely under gravity and first strikes the horizontal surface at the point C. Given that r = 0.4m,

find the distance AC.

A uniform lamina is in the shape of a quadrant of a circle of radius a. Show, by integration, that the centre of mass of the lamina is at a distance of from each of its straight edges.

A second uniform lamina ABCDEFA is shown shaded in Figure 3. The straight sides AC and AE are perpendicular and AC = AE = 2 a. In the figure, the midpoint of AC is B, the midpoint of AE is F, and ABDF and DGEF are squares of side a. BCD is a quadrant of a circle with centre B. DGE is a quadrant of a circle with centre G.

Find the distance of the centre of mass of the lamina from the side AE.

The lamina is smoothly hinged to a horizontal axis which passes through E and is perpendicular to the plane of the lamina. The lamina has weight W newtons. The lamina is held in equilibrium in a vertical plane, with A vertically above E, by a horizontal force of magnitude X newtons applied at C.

Find X in terms of W.

Two points A and B are 4 m apart on a smooth horizontal surface. A light elastic string, of natural length 0.8 m and modulus of elasticity 15 N, has one end attached to the point A. A light elastic string, of natural length 0.8 m and modulus of elasticity 10 N, has one end attached to the point B. A particle P of mass 0.2 kg is attached to the free end of each string. The particle rests in equilibrium on the surface at the point C on the straight line between A and B.

Show that the length of AC is 1.76 m.

The particle P is now held at the point D on the line AB such that 2.16 m. The particle is then released from rest and in the subsequent motion both strings remain taut.

Show that P moves with simple harmonic motion.

Find the speed of P as it passes through the point C.

Find the time from the instant when P is released from D until the instant when P is first moving with speed 2 m s^–1.