SPS SPS FM Mechanics (SPS FM Mechanics) 2025 January

1. A smooth uniform sphere \(A\), of mass \(5 m\) and radius \(r\), is at rest on a smooth horizontal plane. A second smooth uniform sphere \(B\), of mass \(3 m\) and radius \(r\), is moving in a straight line on the plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and strikes \(A\). Immediately before the impact the direction of motion of \(B\) makes an angle of \(60 ^ { \circ }\) with the line of centres of the spheres. The direction of motion of \(B\) is turned through an angle of \(30 ^ { \circ }\) by the impact. Find

1. the speed of \(B\) immediately after the impact,
2. the coefficient of restitution between the spheres.

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(2 a\) and modulus of elasticity 6 mg . The other end of the string is attached to a fixed point \(A\). The particle moves with constant speed \(v\) in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\) and \(O A = 2 a\), as shown in Figure 2.

1. Show that the extension in the string is \(\frac { 2 } { 5 } a\).
2. Find \(v ^ { 2 }\) in terms of \(a\) and \(g\).
   [0pt] [Question 2 Continued]

3. A particle \(P\) of mass \(2 m\) is moving in a straight line with speed \(3 u\) on a smooth horizontal table. A second particle \(Q\) of mass \(3 m\) is moving in the opposite direction to \(P\) along the same straight line with speed \(u\). The particle \(P\) collides directly with \(Q\). The direction of motion of \(P\) is reversed by the collision. The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Show that the speed of \(Q\) immediately after the collision is \(\frac { u } { 5 } ( 8 e + 3 )\)
2. Find the range of possible values of \(e\). The total kinetic energy of the particles before the collision is \(T\). The total kinetic energy of the particles after the collision is \(k T\). Given that \(e = \frac { 1 } { 2 }\)
3. find the value of \(k\).
   [0pt] [Question 3 Continued]

4. One end \(A\) of a light elastic string \(A B\), of modulus of elasticity \(m g\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(A B\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(A B = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.

1. Show that when the particle comes to rest it has moved a distance \(2 a ( \sin \theta - \mu \cos \theta )\) down the plane.
2. Given that there is no further motion, show that \(\mu \geqslant \frac { 1 } { 3 } \tan \theta\).
   [0pt] [Question 4 Continued]

5. \begin{figure}[h] \end{figure} A uniform solid right circular cylinder has height \(h\) and radius \(r\). The centre of one plane face is \(O\) and the centre of the other plane face is \(Y\). A cylindrical hole is made by removing a solid cylinder of radius \(\frac { 1 } { 4 } r\) and height \(\frac { 1 } { 4 } h\) from the end with centre \(O\). The axis of the cylinder removed is parallel to \(O Y\) and meets the end with centre \(O\) at \(X\), where \(O X = \frac { 1 } { 4 } r\). One plane face of the cylinder removed coincides with the plane face through \(O\) of the original cylinder. The resulting solid \(S\) is shown in Figure 3.

1. Show that the centre of mass of \(S\) is at a distance \(\frac { 85 h } { 168 }\) from the plane face
   containing \(O\). containing \(O\). The solid \(S\) is freely suspended from \(O\). In equilibrium the line \(O Y\) is inclined at an angle \(\arctan ( 17 )\) to the horizontal.
2. Find \(r\) in terms of \(h\).
   [0pt] [Question 5 Continued]
   [0pt] [Question 5 Continued]