OCR GCE AS Level Further Maths B (MEI) Mechanics b (withdrawn AS option) June 2019 question paper Y415

A small object of mass 5 kg is attached to one end of each of two identical parallel light elastic strings. The upper ends of both strings are attached to a horizontal ceiling. The object hangs in equilibrium at R, with the extension of each string being 0.1 m, as shown in

Find the stiffness of each string.

One of the strings is now removed and the object initially falls downwards. The object does not return to R at any point in the subsequent motion.

Suggest a reason why the object does not return to R.

A particle P of mass m travels in a straight line on a smooth horizontal surface. At time t, P is a distance x from a fixed point O and is moving with speed v away from O. A horizontal force of magnitude 3 mt acts on P, in a direction away from O.

Verify that the general solution of this differential equation is x = ³ + At + k, where A and k

are constants.

Given that x = 6 and v = 12 when t = 1, find the values of A and k.

Find the equation of the path of Q, giving your answer in the form y = ax² + bx + c, where a

b and c are constants to be determined.

What can you deduce about the path of Q from the value of the constant c you found in

part (b) ?

Two uniform discs, A of mass 0.2 kg and B of mass 0.5 kg, collide with smooth contact while moving on a smooth horizontal surface. Immediately before the collision, A is moving with speed 0.5ms^-1 at an angle α with the line of centres, where sin α = 0.6, and B is moving with speed 0.3ms^-1 at right angles to the line of centres. A straight smooth vertical wall is situated to the right of B, perpendicular to the line of centres, as shown in Fig. 4. The coefficient of restitution between A and B is 0.75.

Find the speeds of A and B immediately after the collision.

Explain why there could be a second collision between A and B if B rebounds from the wall

with sufficient speed.

Find the range of values of the coefficient of restitution between B and the wall for which

there will be a second collision between A and B.

How does your answer to part (b) change if the contact between B and the wall is not smooth?

Fig. 5 shows the curve with equation y =-x² + 4x + 2. The curve intersects the x-axis at P and Q. The region bounded by the curve, the x-axis, the y-axis and the line x = 4 is occupied by a uniform lamina L. The horizontal base of L is OA, where A is the point (4, 0).

Explain why the centre of mass of L lies on the line x = 2.

In this question you must show detailed reasoning.

Find the y-coordinate of the centre of mass of L.

L is freely suspended from A. Find the angle AO makes with the vertical.

The region bounded by the curve and the x-axis is now occupied by a uniform lamina M. The horizontal base of M is PQ.

Explain how the position of the centre of mass of M differs from the position of the centre of

mass of L.

A smooth solid hemisphere of radius a is fixed with its plane face in contact with a horizontal surface. The highest point on the hemisphere is H, and the centre of its base is O. A particle of mass m is held at a point S on the surface of the hemisphere such that angle HOS is 30°, as shown in Fig. 6.

Fig. 6

Show that the particle passes through H without leaving the surface of the hemisphere.

After passing through H, the particle passes through a point Q on the surface of the hemisphere, where angle HOQ = θ °.

State, in terms of g and θ, the tangential component of the acceleration of the particle when it

is at Q.

The particle loses contact with the hemisphere at Q and subsequently lands on the horizontal surface at a point L.

Find the value of cos θ correct to 3 significant figures.

Show that OL = ka, where k is to be found correct to 3 significant figures.