CAIE M2 (Mechanics 2) 2014 November

1 A golf ball \(B\) is projected from a point \(O\) on horizontal ground. \(B\) hits the ground for the first time at a point 48 m away from \(O\) at time 2.4 s after projection. Calculate the angle of projection.

2 A particle \(P\) of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 64 N . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal surface. \(P\) is placed on the surface at a point 0.8 m from \(A\). The particle \(P\) is then projected with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly away from \(A\).

1. Calculate the distance \(A P\) when \(P\) is at instantaneous rest.
2. Calculate the speed of \(P\) when it is 1.0 m from \(A\).

3 A small ball of mass \(m \mathrm {~kg}\) is projected vertically upwards with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards when it is \(x \mathrm {~m}\) above the point of projection. A resisting force of magnitude \(0.02 m v \mathrm {~N}\) acts on the ball during its upward motion.

1. Show that, while the ball is moving upwards, \(\left( \frac { 500 } { v + 500 } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 0.02\).
2. Find the greatest height of the ball above its point of projection.

4 A particle \(P\) is projected with speed \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on a horizontal plane.

1. Calculate the speed of \(P\) when it has been in motion for 4 s , and calculate another time at which \(P\) has this speed.
2. Find the distance \(O P\) when \(P\) has been in motion for 4 s .

5
\includegraphics[max width=\textwidth, alt={}, center]{81be887c-ab01-4327-a5df-f25c68a6fdb6-2_337_517_1749_813} Two light elastic strings each have one end attached to a fixed horizontal beam. One string has natural length 0.6 m and modulus of elasticity 12 N ; the other string has natural length 0.7 m and modulus of elasticity 21 N . The other ends of the strings are attached to a small block \(B\) of weight \(W \mathrm {~N}\). The block hangs in equilibrium \(d \mathrm {~m}\) below the beam, with both strings vertical (see diagram).

1. Given that the tensions in the strings are equal, find \(d\) and \(W\). The small block is now raised vertically to the point 0.7 m below the beam, and then released from rest.
2. Find the greatest speed of the block in its subsequent motion.

6 A horizontal disc with a rough surface rotates about a fixed vertical axis which passes through the centre of the disc. A particle \(P\) of mass 0.2 kg is in contact with the surface and rotates with the disc, without slipping, at a distance 0.5 m from the axis. The greatest speed of \(P\) for which this motion is possible is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the coefficient of friction between the disc and \(P\).
   \(P\) is now attached to one end of a light elastic string, which is connected at its other end to a point on the vertical axis above the disc. The tension in the string is equal to half the weight of \(P\). The disc rotates with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) and \(P\) rotates with the disc without slipping. \(P\) moves in a circle of radius 0.5 m , and the taut string makes an angle of \(30 ^ { \circ }\) with the horizontal.
2. Find the greatest and least values of \(\omega\) for which this motion is possible.
3. Calculate the value of \(\omega\) for which the disc exerts no frictional force on \(P\).

7
\includegraphics[max width=\textwidth, alt={}, center]{81be887c-ab01-4327-a5df-f25c68a6fdb6-3_586_527_1030_810} A uniform lamina \(A B C\) is in the form of a major segment of a circle with centre \(O\) and radius 0.35 m . The straight edge of the lamina is \(A B\), and angle \(A O B = \frac { 2 } { 3 } \pi\) radians (see diagram).

1. Show that the centre of mass of the lamina is 0.0600 m from \(O\), correct to 3 significant figures. The weight of the lamina is 14 N . It is placed on a rough horizontal surface with \(A\) vertically above \(B\) and the lowest point of the arc \(B C\) in contact with the surface. The lamina is held in equilibrium in a vertical plane by a force of magnitude \(F \mathrm {~N}\) acting at \(A\).
2. Find \(F\) in each of the following cases:
   (a) the force of magnitude \(F \mathrm {~N}\) acts along \(A B\);
   (b) the force of magnitude \(F \mathrm {~N}\) acts along the tangent to the circular arc at \(A\).