Edexcel M3 (Mechanics 3) 2005 January

1. A particle \(P\) of mass 0.5 kg is attached to one end of a light inextensible string of length 1.5 m . The other end of the string is attached to a fixed point \(A\). The particle is moving, with the string taut, in a horizontal circle with centre \(O\) vertically below \(A\). The particle is moving with constant angular speed \(2.7 \mathrm { rad } \mathrm { s } ^ { - 1 }\). Find
   1. the tension in the string,
   2. the angle, to the nearest degree, that \(A P\) makes with the downward vertical.
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-2_966_706_841_641}
   \end{figure} A child's toy consists of a uniform solid hemisphere, of mass \(M\) and base radius \(r\), joined to a uniform solid right circular cone of mass \(m\), where \(2 m < M\). The cone has vertex \(O\), base radius \(r\) and height \(3 r\). Its plane face, with diameter \(A B\), coincides with the plane face of the hemisphere, as shown in Figure 1.
2. Show that the distance of the centre of mass of the trom \(A B\) is $$\frac { 3 ( M - 2 m ) } { 8 ( M + m ) } r$$ The toy is placed with \(O A\) on a horizontal surface. The toy is released from rest and does not remain in equilibrium.
3. Show that \(M > 26 m\).

3. \begin{figure}[h] \end{figure} A uniform lamina occupies the region \(R\) bounded by the \(x\)-axis and the curve $$y = \sin x , \quad 0 \leq x \leq \pi$$ as shown in Figure 2.

1. Show, by integration, that the \(y\)-coordinate of the centre of mass of the lamina is \(\frac { \pi } { 8 }\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-3_652_792_1439_568}
   \end{figure} A uniform prism \(S\) has cross-section \(R\). The prism is placed with its rectangular face on a table which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The cross-section \(R\) lies in a vertical plane as shown in Figure 3. The table is sufficiently rough to prevent \(S\) sliding. Given that \(S\) does not topple,
2. find the largest possible value of \(\theta\).
   (3)

4. \begin{figure}[h] \end{figure} In a game at a fair, a small target \(C\) moves horizontally with simple harmonic motion between the points \(A\) and \(B\), where \(A B = 4 L\). The target moves inside a box and takes 3 s to travel from \(A\) to \(B\). A player has to shoot at \(C\), but \(C\) is only visible to the player when it passes a window \(P Q\), where \(P Q = b\). The window is initially placed with \(Q\) at the point as shown in Figure 4. The target \(C\) takes 0.75 s to pass from \(Q\) to \(P\).

1. Show that \(b = ( 2 - \sqrt { 2 } ) L\).
2. Find the speed of \(C\) as it passes \(P\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-4_286_1082_1327_424}
   \end{figure} For advanced players, the window \(P Q\) is moved to the centre of \(A B\) so that \(A P = Q B\), as shown in Figure 5.
3. Find the time, in seconds to 2 decimal places, taken for \(C\) to pass from \(Q\) to \(P\) in this new position.

5. At time \(t = 0\), a particle \(P\) is at the origin \(O\), moving with speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the \(x\)-axis, in the positive \(x\)-direction. At time \(t\) seconds \(( t > 0 )\) the acceleration of \(P\) has magnitude \(\frac { 3 } { \sqrt { } ( t + 4 ) } \mathrm { m } \mathrm { s } ^ { - 2 }\) and is directed towards \(O\).

1. Show that, at time \(t\) seconds, the velocity of \(P\) is \([ 30 - 6 \sqrt { } ( t + 4 ) ] \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find the distance of \(P\) from \(O\) when \(P\) comes to instantaneous rest.

6. A light spring of natural length \(L\) has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The particle is moving vertically. As it passes through the point \(B\) below \(A\), where \(A B = L\), its speed is \(\sqrt { } ( 2 g L )\). The particle comes to instantaneous rest at a point \(C , 4 L\) below \(A\).

1. Show that the modulus of elasticity of the spring is \(\frac { 8 m g } { 9 }\). At the point \(D\) the tension in the spring is \(m g\).
2. Show that \(P\) performs simple harmonic motion with centre \(D\).
3. Find, in terms of \(L\) and \(g\),
   1. the period of the simple harmonic motion,
   2. the maximum speed of \(P\).
      (5)

7. \begin{figure}[h] \end{figure} A trapeze artiste of mass 60 kg is attached to the end \(A\) of a light inextensible rope \(O A\) of length 5 m . The artiste must swing in an arc of a vertical circle, centre \(O\), from a platform \(P\) to another platform \(Q\), where \(P Q\) is horizontal. The other end of the rope is attached to the fixed point \(O\) which lies in the vertical plane containing \(P Q\), with \(\angle P O Q = 120 ^ { \circ }\) and \(O P = O Q = 5 \mathrm {~m}\), as shown in Figure 6. As part of her act, the artiste projects herself from \(P\) with speed \(\sqrt { } 15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the rope \(O A\) and in the plane \(P O Q\). She moves in a circular arc towards \(Q\). At the lowest point of her path she catches a ball of mass \(m \mathrm {~kg}\) which is travelling towards her with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and parallel to \(Q P\). After catching the ball, she comes to rest at the point \(Q\). By modelling the artiste and the ball as particles and ignoring her air resistance, find

1. the speed of the artiste immediately before she catches the ball,
2. the value of \(m\),
3. the tension in the rope immediately after she catches the ball.