SPS SPS FM Mechanics (SPS FM Mechanics) 2023 January

2. \begin{figure}[h] \end{figure} A hockey stick is modelled as a uniform rod \(O A\) of length \(14 r\) joined to a uniform semicircular arc \(A B\) of diameter \(2 r\), as shown in Figure 1. The rod and the arc lie in the same plane and are made of the same material.

1. Find, in terms of \(\pi\) and \(r\), the distance of the centre of mass of the hockey stick from the line \(A B\). The hockey stick is freely suspended from \(O\) and hangs in equilibrium.
   Given that the centre of mass of the hockey stick is a distance \(\frac { \pi r } { ( 14 + \pi ) }\) from \(O A\),
2. find, in degrees to 3 significant figures, the angle between \(O A\) and the vertical.
   [0pt] [Question 2 Continued]

3. \begin{figure}[h] \end{figure} \section*{In this question use \(\boldsymbol { g } = \mathbf { 1 0 m s } \boldsymbol { s } ^ { \mathbf { - 2 } }\).} A light elastic string has natural length \(a\) metres and modulus of elasticity \(\lambda\) newtons. A particle \(P\) of mass 2 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\) on a rough inclined plane. The plane is inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\) The point \(B\) on the plane lies below \(A\) on the line of greatest slope of the plane through \(A\) and \(A B = 3 a\) metres, as shown in Figure 3. The particle \(P\) is held at \(B\) and then released from rest. The particle first comes to instantaneous rest at \(A\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\)

1. Show that \(\lambda = 24\)
2. Find the magnitude of the acceleration of \(P\) at the instant it is released from \(B\).
3. Explain why the answer to part (b) is the greatest value of the magnitude of the acceleration of \(P\) as \(P\) moves from \(B\) to \(A\).
   [0pt] [Question 3 Continued] \section*{4.}

4. A uniform ladder \(A B\) of length 5 m and mass 8 kg is placed at an angle \(\theta\) to the horizontal, with \(A\) on rough horizontal ground and \(B\) against a smooth vertical wall. The coefficient of friction between the ladder and the ground is 0.4 .

1. By taking moments, find the smallest value of \(\theta\) for which the ladder is in equilibrium.
2. A man of mass 75 kg stands on the ladder when \(\theta = 60 ^ { \circ }\). Find the greatest distance from \(A\) that he can stand without the ladder slipping.
   [0pt] [Question 4 Continued]

5. Two smooth circular discs \(A\) and \(B\) are moving on a horizontal plane. The masses of \(A\) and \(B\) are 3 kg and 4 kg respectively. At the instant before they collide

• the velocity of \(A\) is \(2 \mathrm {~ms} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the line joining their centres,
• the velocity of \(B\) is \(5 \mathrm {~ms} ^ { - 1 }\) towards \(A\) along the line joining their centres (see Fig. 6).

\begin{figure}[h] \end{figure} Given that the velocity of \(A\) after the collision is perpendicular to the velocity of \(A\) before the collision find the coefficient of restitution between \(A\) and \(B\).
[0pt] [Question 5 Continued] \section*{6.}

6. A light elastic string of natural length \(a\) has modulus of elasticity \(k m g\), where \(k\) is a constant. One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle of mass \(m\). The particle moves, with the string stretched, in a horizontal circle with constant angular speed \(\omega\), with the centre of the circle vertically below \(O\).

1. Show that, if the string makes a constant angle \(\theta\) with the vertical, $$\cos \theta = \frac { k g - a \omega ^ { 2 } } { k a \omega ^ { 2 } }$$
2. Show that \(\omega < \sqrt { \frac { k g } { a } }\)
   [0pt] [Question 6 Continued] Spare space for extra working Spare space for extra working Spare space for extra working Spare space for extra working
   [0pt] [End of Examination]