CAIE M2 (Mechanics 2) 2016 November

1 A stone \(S\) is thrown horizontally from the top \(T\) of a high tower. At the instant 1.6 s after \(S\) is thrown, the line \(S T\) makes an angle of \(30 ^ { \circ }\) below the horizontal. Find the speed with which \(S\) is thrown. [3]

2 A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string with modulus of elasticity 24 N and natural length 0.6 m . The other end of the string is attached to a fixed point \(A\). The particle \(P\) hangs in equilibrium vertically below \(A\).

1. Find the distance \(A P\). The particle \(P\) is raised to \(A\) and released from rest.
2. Calculate the greatest speed of \(P\) in the subsequent motion.

3
\includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-2_442_789_941_676} A non-uniform \(\operatorname { rod } A B\) of length 0.5 m is freely hinged to a fixed point at \(A\). The rod is in equilibrium at an angle of \(30 ^ { \circ }\) with the horizontal with \(B\) below the level of \(A\). Equilibrium is maintained by a force of magnitude \(F\) N applied at \(B\) acting at \(45 ^ { \circ }\) above the horizontal in the vertical plane containing \(A B\). The force exerted by the hinge on the rod has magnitude 10 N and acts at an angle of \(60 ^ { \circ }\) above the horizontal (see diagram).

1. By resolving horizontally and vertically, calculate \(F\) and the weight of the rod.
2. Find the distance of the centre of mass of the rod from \(A\).

4 A particle \(P\) is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground. \(P\) subsequently bounces when it first strikes the ground at the point \(A\).

1. Find the time after projection when \(P\) first strikes the ground, and the distance \(O A\). When \(P\) bounces at \(A\) the horizontal component of the velocity of \(P\) is unchanged. The vertical component of velocity is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) immediately after bouncing. \(P\) strikes the ground for the second time at \(B\) where it remains at rest.
2. Calculate the first and last times after projection at which the speed of \(P\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

5 A particle \(P\) of mass 0.4 kg is released from rest at a point \(O\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A force of magnitude \(3 \mathrm { e } ^ { - t } \mathrm {~N}\) directed up a line of greatest slope acts on \(P\), where \(t \mathrm {~s}\) is the time after release.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 7.5 \mathrm { e } ^ { - t } - 5\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) up the plane at time \(t \mathrm {~s}\).
2. Express \(v\) in terms of \(t\).
3. Find the distance of \(P\) from \(O\) when \(v\) has its maximum value.

6
\includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-3_656_757_781_694} The diagram shows the cross-section \(A B C D E F\) through the centre of mass of a uniform prism which rests with \(A B\) on rough horizontal ground. \(A B C D\) is a rectangle with \(A B = C D = 0.4 \mathrm {~m}\) and \(B C = A D = 1.8 \mathrm {~m}\). The other part of the cross-section is a semicircle with diameter \(D F\) and radius \(r \mathrm {~m}\).

1. Given that the prism is on the point of toppling, show that \(r = 0.6\). A force of magnitude \(P \mathrm {~N}\) is applied to the prism, acting at \(60 ^ { \circ }\) to the upwards vertical along a tangent to the semicircle at a point between \(D\) and \(E\). The prism has weight 15 N and is in equilibrium on the point of toppling about \(B\).
2. Show that \(P = 3.26\), correct to 3 significant figures.
3. Find the smallest possible value of the coefficient of friction between the prism and the ground.

7
\includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-4_213_811_260_667} A small ball \(B\) of mass 0.5 kg moves in a horizontal circle with centre \(O\) and radius 0.4 m on the smooth inner surface of a hollow cone fixed with its vertex down. The axis of the cone is vertical and the semi-vertical angle is \(60 ^ { \circ }\) (see diagram).

1. Show that the magnitude of the force exerted by the cone on \(B\) is 5.77 N , correct to 3 significant figures, and calculate the angular speed of \(B\). One end of a light elastic string of natural length 0.45 m and modulus of elasticity 36 N is attached to \(B\). The other end of the string is attached to the point on the axis 0.3 m above \(O\). The ball \(B\) again moves on the surface of the cone in the same horizontal circle as before.
2. Calculate the speed of \(B\).