Edexcel A Level Further Maths M3 · Mechanics 3 June 2015 question paper 6679

A particle P of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity newtons. The other end of the spring is attached to a fixed point A on a ceiling. The particle is hanging freely in equilibrium at a distance 1.5 m vertically below A.

Find the value of .

The particle is now raised to the point B, where B is vertically below A and AB = 0.8 m. The spring remains straight. The particle is released from rest and first comes to instantaneous rest at the point C.

Find the distance AC.

The finite region bounded by the x-axis, the curve with equation y = 2e^x, the y-axis and the line x = 1 is rotated through one complete revolution about the x-axis to form a uniform solid. Use algebraic integration to

show that the volume of the solid is 2(e² – 1),

find, in terms of e, the x coordinate of the centre of mass of the solid.

A small ball P of mass m is attached to the midpoint of a light inextensible string of length 4 l. The ends of the string are attached to fixed points A and B, where A is vertically above B. Both strings are taut and AP makes an angle of 30° with AB, as shown in Figure 1. The ball is moving in a horizontal circle with constant angular speed .

Find, in terms of m, g, l and ,

the tension in AP,

the tension in BP.

A vehicle of mass 900 kg moves along a straight horizontal road. At time t seconds the resultant force acting on the vehicle has magnitude N, where k is a positive constant. The force acts in the direction of motion of the vehicle. At time t seconds, t 1, the speed of the vehicle is v m s^–1 and the vehicle is a distance x metres from a fixed point O on the road. When t = 1 the vehicle is at rest at O and when t = 4 the speed of the vehicle is 10.5 m s^–1.

Hence deduce that the speed of the vehicle never reaches 14 m s^–1.

Use the trapezium rule, with 4 equal intervals, to estimate the value of x when v = 7

Figure 2 shows a uniform solid spindle which is made by joining together the circular faces of two right circular cones. The common circular face has radius r and centre O. The smaller cone has height h and the larger cone has height kh. The point A lies on the circumference of the common circular face. The spindle is suspended from A and hangs freely in equilibrium with AO at an angle of 30° to the vertical.

Two points A and B are 6 m apart on a smooth horizontal floor. A particle P of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2.5 m and modulus of elasticity 20 N. The other end of the spring is attached to A. A second light elastic spring, of natural length 1.5 m and modulus of elasticity 18 N, has one end attached to P and the other end attached to B, as shown in Figure 3. Initially P rests in equilibrium at the point O, where AOB is a straight line.

Find the length of AO.

The particle P now receives an impulse of magnitude 6 N s acting in the direction OB and P starts to move towards B.

Show that P moves with simple harmonic motion about O.

Find the amplitude of the motion.

Find the time taken by P to travel 1.2 m from O.

A solid smooth sphere, with centre O and radius r, is fixed to a point A on a horizontal floor. A particle P is placed on the surface of the sphere at the point B, where B is vertically

on the surface of the sphere. When OP makes an angle with the upward vertical and P remains in contact with the sphere, the speed of P is v.

The particle leaves the surface of the sphere when θ =

Find the value of cos.

After leaving the surface of the sphere, P moves freely under gravity and hits the floor at the point C. Given that r = 0.5m,

find, to 2 significant figures, the distance AC.