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

A particle P is moving along the positive x-axis. When the displacement of P from the origin is x metres, the velocity of P is v m s^–1 and the acceleration of P is 9x m s^–2. When x = 2, v = 6 Show that v² = 9x².

A uniform solid consists of a right circular cone of radius r and height kr, where k > fixed to a hemisphere of radius r. The centre of the plane face of the hemisphere is O and this plane face coincides with the base of the cone, as shown in Figure 1.

Show that the distance of the centre of mass of the solid from O is

The point A lies on the circumference of the base of the cone. The solid is suspended by a string attached at A and hangs freely in equilibrium. The angle between AO and the

Find the value of k.

A particle P of mass 0.6 kg is moving along the x-axis in the positive direction. At time t = 0, P passes through the origin O with speed 15 m s^–1. At time t seconds the distance OP is x metres, the speed of P is v m s^–1 and the resultant force acting on P has magnitude newtons. The resultant force is directed towards O.

Find the value of x when t = 5

A particle P of mass m is attached to one end of a light elastic string, of natural length 2 a and modulus of elasticity 6 mg. The other end of the string is attached to a fixed point A. The particle moves with constant speed v in a horizontal circle with centre O, where O is vertically below A and OA = 2 a, as shown in Figure 2.

Show that the extension in the string is a.

Find v² in terms of a and g.

A particle P is moving in a straight line with simple harmonic motion on a smooth horizontal floor. The particle comes to instantaneous rest at points A and B where AB is 0.5 m. The mid-point of AB is O. The mid-point of OA is C. The mid-point of OB is D. The particle takes 0.2 s to travel directly from C to D. At time t = 0, P is moving through O towards A.

Show that the period of the motion is s.

Find the distance of P from B when t = 2 s.

Find the maximum magnitude of the acceleration of P.

Find the maximum speed of P.

A smooth hollow cylinder of internal radius a is fixed with its axis horizontal. A particle P moves on the inner surface of the cylinder in a vertical circle with radius a and centre O, where O lies on the axis of the cylinder. The particle is projected vertically downwards with speed u from point A on the circle, where OA is horizontal. The particle first loses contact with the cylinder at the point B, where AOB = 150, as shown in Figure 3. Given that air resistance can be ignored,

show that the speed of P at B is ,

find u in terms of a and g.

After losing contact with the cylinder, P crosses the diameter through A at the point D. At D the velocity of P makes an angle with the horizontal.

A particle P of mass 1.5 kg is attached to the mid-point of a light elastic string of natural length 0.30 m and modulus of elasticity newtons. The ends of the string are attached to two fixed points A and B, where AB is horizontal and AB = 0.48 m. Initially P is held at rest at the mid-point, M, of the line AB and the tension in the string is 240 N.

Show that = 400

The particle is now held at rest at the point C, where C is 0.07 m vertically below M. The particle is released from rest at C.

Find the magnitude of the initial acceleration of P.

Find the speed of P as it passes through M.