SPS SPS FM Mechanics (SPS FM Mechanics) 2024 January

3.
\includegraphics[max width=\textwidth, alt={}, center]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-06_397_878_159_571} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 4 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The spheres are moving in opposite directions, each at \(60 ^ { \circ }\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.

1. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres.
2. Find the coefficient of restitution between the spheres.
   [0pt] [Question 3 Continued]

4. A uniform heavy lamina occupies the region shaded in Fig. 3. This region is formed by removing a square of side 1 unit from a square of side \(a\) units (where \(a > 1\) ). \begin{figure}[h] \end{figure} Relative to the axes shown in Fig. 3, the centre of mass of the lamina is at \(( \bar { x } , \bar { y } )\).

1. Show that \(\bar { x } = \bar { y } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }\).
   [0pt] [You may need to use the result \(\frac { a ^ { 3 } - 1 } { 2 \left( a ^ { 2 } - 1 \right) } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }\).]
2. Show that the centre of mass of the lamina lies on its perimeter if \(a = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\).
3. With the value of \(a = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\) the lamina is suspended from A and hangs in equilibrium. Find the angle that the line OA makes with the vertical.
   [0pt] [Question 4 Continued] \section*{5.}

5. A cone of semi-vertical angle \(60 ^ { \circ }\) is fixed with its axis vertical and vertex upwards. A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. The particle moves in a horizontal circle on the smooth outer surface of the cone with constant angular speed \(\omega\), with the string making a constant angle \(60 ^ { \circ }\) with the horizontal, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5f9a87c6-2255-4178-ab04-441bb0cc4ce0-10_538_648_456_664}

1. Find the tension in the string, in terms of \(m , l , \omega\) and \(g\). The particle remains on the surface of the cone.
2. Show that the time for the particle to make one complete revolution is greater than $$2 \pi \sqrt { \frac { l \sqrt { 3 } } { 2 g } } .$$ [Question 5 Continued]
   [0pt] [Question 5 Continued]