Edexcel A Level Further Maths M3 · Mechanics 3 June 2008 question paper 6679

A light elastic spring, of natural length L and modulus of elasticity , has a particle P of mass m attached to one end. The other end of the spring is fixed to a point O on the closed end of a fixed smooth hollow tube of length L. The tube is placed horizontally and P is held inside the tube with OP = L, as shown in Figure 1. The particle P is released and passes through the open end of the tube with speed √(2gL).

Show that .= 8mg

The tube is now fixed vertically and P is held inside the tube with OP = L and P above O. The particle P is released and passes through the open top of the tube with speed u.

Find u.

A particle P moves with simple harmonic motion and comes to rest at two points A and B which are 0.24 m apart on a horizontal line. The time for P to travel from A to B is 1.5 s. The midpoint of AB is O. At time t = 0, P is moving through O, towards A, with speed u m s^–1.

Find the value of u.

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

Find the speed of P when t = 2 s.

Figure 2 shows a particle B, of mass m, attached to one end of a light elastic string. The other end of the string is attached to a fixed point A, at a distance h vertically above a smooth horizontal table. The particle moves on the table in a horizontal circle with centre O, where O is vertically below A. The string makes a constant angle θ with the downward vertical and B moves with constant angular speed ω about OA.

The elastic string has natural length h and modulus of elasticity 2 mg.

find ω in terms of g and h.

A uniform solid hemisphere, of radius 6 a and centre O, has a solid hemisphere of radius 2 a, and centre O, removed to form a bowl B as shown in Figure 3.

Show that the centre of mass of B is from aa O.

The bowl B is fixed to a plane face of a uniform solid cylinder made from the same material as B. The cylinder has radius 2 a and height 6 a and the combined solid S has an axis of symmetry which passes through O, as shown in Figure 4.

Show that the centre of mass of S a from O. The plane surface of the cylindrical base of S is placed on a rough plane inclined at 12° to the horizontal. The plane is sufficiently rough to prevent slipping.

Determine whether or not S will topple.

A particle P of mass m is attached to one end of a light inextensible string of length a. The other end of the string is attached to a fixed point O. The particle is released from rest with the string taut and OP horizontal.

Find the tension in the string when OP makes an angle of 60° with the downward vertical.

A particle Q of mass 3 m is at rest at a distance a vertically below O. When P strikes Q the particles join together and the combined particle of mass 4 m starts to move in a vertical circle with initial speed u.

The combined particle comes to instantaneous rest at A.

Find

the angle that the string makes with the downward vertical when the combined particle is at A,

the tension in the string when the combined particle is at A.

A particle P of mass 0.5 kg moves along the positive x-axis. It moves away from the origin O under the action of a single force directed away from O. When OP = x metres, the magnitude of the force is N and the speed of P is v m s^–1. Initially P is at rest at O.

Show that the speed of P never reaches √6 m s^–1.

Find x when P has been moving for 2 seconds.