Edexcel GCE Mechanics M2 January 2011 exam answers review

A cyclist starts from rest and moves along a straight horizontal road. The combined mass of the cyclist and his cycle is 120 kg. The resistance to motion is modelled as a constant force of magnitude 32 N. The rate at which the cyclist works is 384 W. The cyclist accelerates until he reaches a constant speed of v m s^–1.

Find

1. the value of v,(3)
2. the acceleration of the cyclist at the instant when the speed is 9 m s^–1.(3)

A particle of mass 2 kg is moving with velocity (5i + j) m s^–1 when it receives an impulse of (–6i + 8j) N s. Find the kinetic energy of the particle immediately after receiving the impulse.

A particle moves along the x-axis. At time t = 0 the particle passes through the origin with speed 8 m s–1 in the positive x-direction. The acceleration of the particle at time t seconds, t≥0, is (4t3 – 12t) m s^–2 in the positive x-direction.

Find

1. the velocity of the particle at time t seconds,(3)
2. the displacement of the particle from the origin at time t seconds,(2)
3. the values of t at which the particle is instantaneously at rest.(3)

A box of mass 30 kg is held at rest at point A on a rough inclined plane. The plane is inclined at 20° to the horizontal. Point B is 50 m from A up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from A to B by a force acting parallel to AB and then held at rest at B. The coefficient of friction between the box and the plane is 1/4. Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,

1. find the work done in dragging the box from A to B.(6)

The box is released from rest at the point B and slides down the slope. Using the workenergy principle, or otherwise,

2. find the speed of the box as it reaches A.(5)

The uniform L-shaped lamina ABCDEF, shown in Figure 2, has sides AB and FE parallel, and sides BC and ED parallel. The pairs of parallel sides are 9 cm apart. The points A, F, D and C lie on a straight line.

AB = BC = 36 cm, FE = ED = 18 cm. ∠ABC = ∠FED = 90°, and ∠BCD = ∠EDF = ∠EFD = ∠BAC = 45°.

1. Find the distance of the centre of mass of the lamina from
   • side AB,
   • side BC.

   (7)

The lamina is freely suspended from A and hangs in equilibrium.

2. Find, to the nearest degree, the size of the angle between AB and the vertical.(3)

At time t = 0, a particle P is projected from the point A which has position vector 10j metres with respect to a fixed origin O at ground level. The ground is horizontal. The velocity of projection of P is (3i + 5j) m s^–1, as shown in Figure 3. The particle moves freely under gravity and reaches the ground after T seconds.

1. For 0≤t≤T, show that, with respect to O, the position vector, r metres, of P at time
   t seconds is given by
   r = 3ti + (10 + 5t – 4.9t2)j

   (3)

2. Find the value of T.(3)
3. Find the velocity of P at time t seconds (0≤t≤T).

When P is at the point B, the direction of motion of P is 45° below the horizontal.

4. Find the time taken for P to move from A to B.(2)
5. Find the speed of P as it passes through B.(2)

A uniform plank AB, of weight 100 N and length 4 m, rests in equilibrium with the end A
on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is
fixed to the ground and cannot move. The point of contact between the plank and the drum
is C, where AC = 3 m, as shown in Figure 4. The plank is resting in a vertical plane which
is perpendicular to the axis of the drum, at an angle θ to the horizontal, where sin θ = 1/3.

The coefficient of friction between the plank and the ground is µ. Modelling the plank as
a rod, find the least possible value of µ.

A particle P of mass m kg is moving with speed 6 m s^–1 in a straight line on a smooth horizontal floor. The particle strikes a fixed smooth vertical wall at right angles and rebounds. The kinetic energy lost in the impact is 64 J. The coefficient of restitution between P and the wall is 1/3.

1. Show that m = 4.

After rebounding from the wall, P collides directly with a particle Q which is moving towards P with speed 3 m s–1. The mass of Q is 2 kg and the coefficient of restitution between P and Q is 1/3

2. Show that there will be a second collision between P and the wall.