CAIE M2 (Mechanics 2) 2012 June

1 The end \(A\) of a \(\operatorname { rod } A B\) of length 1.2 m is freely pivoted at a fixed point. The rod rotates about \(A\) in a vertical plane. Calculate the angular speed of the rod at an instant when \(B\) has speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2
\includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-2_481_412_440_865} The diagram shows a circular object formed from a uniform semicircular lamina of weight 11 N and a uniform semicircular arc of weight 9 N . The lamina and the arc both have centre \(O\) and radius 0.7 m and are joined at the ends of their common diameter \(A B\).

1. Show that the distance of the centre of mass of the object from \(O\) is 0.0371 m , correct to 3 significant figures. The object hangs in equilibrium, freely suspended at \(A\).
2. Find the angle between \(A B\) and the vertical and state whether the lowest point of the object is on the lamina or on the arc.

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small sphere \(S\) of mass \(m \mathrm {~kg}\) is moving inside a smooth hollow bowl whose axis is vertical and whose sloping side is inclined at \(60 ^ { \circ }\) to the horizontal. \(S\) moves with constant speed in a horizontal circle of radius 0.6 m (see Fig. 1). \(S\) is in contact with both the plane base and the sloping side of the bowl (see Fig. 2).

1. Given that the magnitudes of the forces exerted on \(S\) by the base and sloping side of the bowl are equal, calculate the speed of \(S\).
2. Given instead that \(S\) is on the point of losing contact with one of the surfaces, find the angular speed of \(S\).

4 A light elastic string has natural length 2.4 m and modulus of elasticity 21 N . A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the mid-point of the string. The ends of the string are attached to fixed points \(A\) and \(B\) which are 2.4 m apart at the same horizontal level. \(P\) is projected vertically upwards with velocity \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the mid-point of \(A B\). In the subsequent motion \(P\) is at instantaneous rest at a point 1.6 m above \(A B\).

1. Find \(m\).
2. Calculate the acceleration of \(P\) when it first passes through a point 0.5 m below \(A B\).

5 A particle \(P\) of mass 0.4 kg is released from rest at the top of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The motion of \(P\) down the slope is opposed by a force of magnitude \(0.6 x \mathrm {~N}\), where \(x \mathrm {~m}\) is the distance \(P\) has travelled down the slope. \(P\) comes to rest before reaching the foot of the slope. Calculate

1. the greatest speed of \(P\) during its motion,
2. the distance travelled by \(P\) during its motion.

6
\includegraphics[max width=\textwidth, alt={}, center]{6d3892e0-8c88-44ec-940f-c526d71a7fc6-3_720_723_1165_712} The diagram shows the cross-section \(O A B C D E\) through the centre of mass of a uniform prism. The interior angles of the cross-section at \(O , A , C , D\) and \(E\) are all right angles. \(O A = 0.4 \mathrm {~m} , A B = 0.5 \mathrm {~m}\) and \(B C = C D = 1 \mathrm {~m}\).

1. Calculate the distance of the centre of mass of the prism from \(O E\). The weight of the prism is 120 N . A force of magnitude \(F \mathrm {~N}\) acting along \(D E\) holds the prism in equilibrium when \(O A\) rests on a rough horizontal surface.
2. Find the set of possible values of \(F\).

7 A small ball \(B\) is projected with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(41 ^ { \circ }\) above the horizontal from a point \(O\) which is 1.6 m above horizontal ground. At time \(t \mathrm {~s}\) after projection the horizontal and vertically upward displacements of \(B\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.

1. Express \(x\) and \(y\) in terms of \(t\) and hence show that the equation of the trajectory of \(B\) is $$y = 0.869 x - 0.0390 x ^ { 2 }$$ where the coefficients are correct to 3 significant figures. A vertical fence is 1.5 m from \(O\) and perpendicular to the plane in which \(B\) moves. \(B\) just passes over the fence and subsequently strikes the ground at the point \(A\).
2. Calculate the height of the fence, and the distance from the fence to \(A\).