Edexcel A Level Further Maths M3 · Mechanics 3 June 2016 question paper 6679

A particle P of mass 0.5kg is moving along the positive x-axis under the action of a resultant force. The force acts along the x-axis. At time t seconds, P is x metres from the origin O and is moving away from O in the positive x direction with speed m s¹

Find the magnitude of the force acting on P when x = 3

Given that x = 4 when t = 2

find the value of t when x = 10

Figure 1 shows a uniform triangular lamina ABC in which AB = 6cm, BC = 9cm and angle ABC = 90°. The centre of mass of the lamina is G. Use algebraic integration to find the distance of G from AB.

One end of a light elastic string, of natural length 1.5m and modulus of elasticity 14.7N, is attached to a fixed point O on a ceiling. A particle P of mass 0.6kg is attached to the free end of the string. The particle is held at O and released from rest. The particle comes to instantaneous rest for the first time at the point A. Find

the distance OA,

the magnitude of the instantaneous acceleration of P at A.

A uniform solid S consists of two right circular cones of base radius r. The smaller cone has height 2 h and the centre of the plane face of this cone is O. The larger cone has height kh where k > 2. The two cones are joined so that their plane faces coincide, as shown in Figure 2.

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

The point A lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at A and hangs freely in equilibrium. Given that r = 3 h and k = 6

find the size of the angle between AO and the vertical.

A particle P of mass m is attached to the ends of two light inextensible strings. The other ends of the strings are attached to fixed points A and B, where B is vertically below A and AB = l. The particle is moving with constant angular speed in a horizontal circle. Both strings are taut and inclined at 30° to AB, as shown in Figure 3.

Find the tension in BP.

Show that the time taken by P to complete one revolution is less

One end of a light inextensible string of length l is attached to a particle P of mass 2 m. The other end of the string is attached to a fixed point A. The particle is hanging freely at

rest with the string vertical. The particle is then projected horizontally with speed

Find the speed of P at the instant when the string is horizontal.

When the string is horizontal and P is moving upwards, the string comes into contact with a small smooth peg which is fixed at the point B, where AB is horizontal and AB l. The particle then describes a complete semicircle with centre B.

A particle P of mass 0.5kg is attached to one end of a light elastic spring, of natural length 1.2m and modulus of elasticity 15N. The other end of the spring is attached to a fixed point A on a smooth horizontal table. The particle is placed on the table at the point B where AB = 1.2m. The particle is pulled away from B to the point C, where ABC is a straight line and BC = 0.8m, and is then released from rest.

Show that P moves with simple harmonic motion with centre B.

Find the period of this motion.

Find the speed of P when it reaches B.

The point D is the midpoint of AB.

Find the time taken for P to move directly from C to D.

When P first comes to instantaneous rest a particle Q of mass 0.3kg is placed at B. When P reaches B again, P strikes and adheres to Q to form a single particle R.

Show that R also moves with simple harmonic motion.

Find the amplitude of this motion.