Edexcel A Level Further Maths M3 · Mechanics 3 June 2017 question paper 6679

A uniform lamina is in the shape of the region R. Region R is bounded by the curve with equation y = 4 – x², the positive x-axis and the positive y-axis, as shown shaded in Figure 1. Use algebraic integration to find the x coordinate of the centre of mass of the lamina.

A particle P of mass m is attached to one end of a light inextensible string. The other end of the string is attached xed point A. The particle moves in a horizontal circle with

the string makes a constant angle ° with the downward vertical, as shown in Figure 2. Given that the tension in the string is 1.2 mg, find

the length of the string.

A particle P of mass m kg is initially held at rest at the point O on a smooth plane which is inclined at 30° to the horizontal. The particle is released from rest and slides down the plane against a force of magnitude mx newtons acting towards O, where x metres is the distance of P from O.

Find the speed of P when x = 3

Find the distance P has moved when it first comes to instantaneous rest.

A thin uniform right hollow cylinder, of radius a and height 4 a, has a base but no top. A thin uniform hemispherical shell, also of radius a, is made of the same material as the cylinder. The hemispherical shell is attached to the open end of the cylinder forming a container C. The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of C is O, as shown in Figure 3.

Find the distance from O to the centre of mass of C.

The container is placed with its circular base on a plane which is inclined at θ° to the horizontal. The plane is sufficiently rough to prevent C from sliding. The container is on the point of toppling.

A hollow cylinder is fixed with its axis horizontal. A particle P moves in a vertical circle, with centre O and radius a, on the smooth inner surface of the cylinder. The particle moves in a vertical plane which is perpendicular to the axis of the cylinder. The particle

horizontal and OA = a. When angle AOP = , the speed of P is v, as shown in Figure 4.

Show that v² = ag (7 + 2sin)

Verify that P will move in a complete circle.

Find the maximum value of v.

The ends of a light elastic string, of natural length 0.4 m and modulus of elasticity λ newtons, are attached to two fixed points A and B which are 0.6 m apart on a smooth horizontal table. The tension in the string is 8 N.

Show that = 16

A particle P is attached to the midpoint of the string. The particle P is now pulled horizontally in a direction perpendicular to AB to a point 0.4 m from the midpoint of AB. The particle is held at rest by a horizontal force of magnitude F newtons acting in a direction perpendicular to AB, as shown in Figure 5 below.

Find the value of F.

The particle is released from rest. Given that the mass of P is 0.3 kg,

find the speed of P as it crosses the line AB.

The fixed points A and B are 4 m apart on a smooth horizontal floor. One end of a light elastic string, of natural length 1.8 m and modulus of elasticity 45 N, is attached to a particle P and the other end is attached to A. One end of another light elastic string, of natural length 1.2 m and modulus of elasticity 20 N, is attached to P and the other end is attached to B. The particle P rests in equilibrium at the point O, where AOB is a straight line, as shown in Figure 6.

Show that AO = 2.2 m.

The point C lies on the straight line AOB with AC = 2.7 m. The mass of P is 0.6 kg. The particle P is held at C and then released from rest.

Show that, while both strings are taut, P moves with simple harmonic motion with centre O.

The point D lies on the straight line AOB with AD = 1.8 m. When P reaches D the string PB breaks.

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