Edexcel A Level Further Maths M3 · Mechanics 3 June 2011 question paper 6679

A particle P of mass 0.5 kg moves on the positive x-axis under the action of a single force directed towards the origin O. At time t seconds the distance of P from O is x metres, the magnitude of the force is 0.375x² N and the speed of P is v m s^–1. When t = 0, OP = 8 m and P is moving towards O with speed 2 m s^–1.

Find the distance of P from O at the instant when v = 5.

The shaded region R is bounded by the curve with equation y = 9 – x², the positive x-axis and the positive y-axis, as shown in Figure 1. A uniform solid S is formed by rotating R through 360° about the x-axis. Find the x-coordinate of the centre of mass of S.

A solid consists of a uniform solid right cylinder of height 5 l and radius 3 l joined to a uniform solid hemisphere of radius 3 l. The plane face of the hemisphere coincides with a circular end of the cylinder and has centre O, as shown in Figure 2. The density of the hemisphere is twice the density of the cylinder.

Find the distance of the centre of mass of the solid from O.

The solid is now placed with its circular face on a plane inclined at an angle ° to the horizontal, as shown in Figure 3. The plane is sufficiently rough to prevent the solid slipping. The solid is on the point of toppling.

A light inextensible string has its ends attached to two fixed points A and B. The point A is vertically above B and AB = 7 a. A particle P of mass m is fixed to the string and moves in a horizontal circle of radius 3a with angular speed . The centre of the circle is C where C lies on AB and AC = 4 a, as shown in Figure 4. Both parts of the string are taut.

Show that the tension in AP is m3a² + g.

Find the tension in BP.

A particle P of mass m is attached to one end of a light elastic string of natural length l and modulus of elasticity 3 mg. The other end of the string is attached to a fixed point O on a rough horizontal table. The particle lies at rest at the point A on the table, where OA . The coefficient of friction between P and the table is .

The particle is now moved along the table to the point B, where OB , and released from rest. Given that , find

the speed of P at the instant when the string becomes slack,

the total distance moved by P before it comes to rest again.

A particle P 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 held at the point A, where OA = a and OA is horizontal. The point B is vertically above O and the point C is vertically below O, with OB = OC = a, as shown in Figure 5. The particle is projected vertically upwards with speed 3˲G ag.

Show that P will pass through B.

Find the speed of P as it reaches C.

As P passes through C it receives an impulse. Immediately after this, the speed of P is ˲G11ag and the direction of motion of P is unchanged.

Find the angle between the string and the downward vertical when P comes to instantaneous rest.

A particle P of mass 0.5 kg is attached to the mid-point of a light elastic string of natural length 1.4 m and modulus of elasticity 2 N. The ends of the string are attached to the points A and B on a smooth horizontal table, where AB = 2 m. The mid-point of AB is O and the point C is on the table between O and B where OC = 0.2 m. At time t = 0 the particle is released from rest at C. At time t seconds the length of the string AP is 1+ x m.

Show that the tension in BP is 3 −10x N.

Find, in terms of x, the tension in AP.

Find the greatest speed of P during the motion.

The point D lies between O and A, where OD = 0.1 m.

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