OCR M3 (Mechanics 3) 2008 January

1 A smooth horizontal surface lies in the \(x - y\) plane. A particle \(P\) of mass 0.5 kg is moving on the surface with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the \(x\)-direction when it is struck by a horizontal blow whose impulse has components - 3.5 N s and 2.4 N s in the \(x\)-direction and \(y\)-direction respectively.

1. Find the components in the \(x\)-direction and the \(y\)-direction of the velocity of \(P\) immediately after the blow. Hence show that the speed of \(P\) immediately after the blow is \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   \(P\) is struck by a second horizontal blow whose impulse is \(\mathbf { I }\).
2. Given that \(P\) 's direction of motion immediately after this blow is parallel to the \(x\)-axis, write down the component of \(\mathbf { I }\) in the \(y\)-direction.

2
\includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-2_515_1065_861_541} Two uniform rods \(A B\) and \(B C\), each of length 2 m , are freely jointed at \(B\). The weights of the rods are \(W \mathrm {~N}\) and 50 N respectively. The end \(A\) of \(A B\) is hinged at a fixed point. The rods \(A B\) and \(B C\) make angles \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)\) and \(\beta\) respectively with the downward vertical, and are held in equilibrium in a vertical plane by a horizontal force of magnitude 75 N acting at \(C\) (see diagram).

1. By taking moments about \(B\) for \(B C\), show that \(\tan \beta = 3\).
2. Write down the horizontal and vertical components of the force acting on \(A B\) at \(B\).
3. Find the value of \(W\).

3
\includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-3_419_921_267_612} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 6 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision the velocity of \(A\) has components \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(B\), and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres. \(B\) is moving with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres towards \(A\) (see diagram). The coefficient of restitution between the spheres is \(e\).

1. Find, in terms of \(e\), the component of the velocity of \(A\) along the line of centres immediately after the collision.
2. Given that the speeds of \(A\) and \(B\) are the same immediately after the collision, and that \(3 e ^ { 2 } = 1\), find \(v\).

4 A particle of mass \(m \mathrm {~kg}\) is released from rest at a fixed point \(O\) and falls vertically. The particle is subject to an upward resisting force of magnitude \(0.49 m v \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the particle when it has fallen a distance of \(x \mathrm {~m}\) from \(O\).

1. Write down a differential equation for the motion of the particle, and show that the equation can be written as \(\left( \frac { 20 } { 20 - v } - 1 \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 0.49\).
2. Hence find an expression for \(x\) in terms of \(v\).

5 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 1.2 m and modulus of elasticity 0.75 mgN . The other end of the string is attached to a fixed point \(O\) of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is released from rest at \(O\) and moves down the plane.

1. Show that the maximum speed of \(P\) is reached when the extension of the string is 0.8 m .
2. Find the maximum speed of \(P\).
3. Find the maximum displacement of \(P\) from \(O\).

6
\includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-4_497_524_276_804} A particle \(P\) of mass 0.4 kg is attached to one end of a light inextensible string of length 2 m . The other end of the string is attached to a fixed point \(O\). With the string taut the particle is travelling in a circular path in a vertical plane. The angle between the string and the downward vertical is \(\theta ^ { \circ }\) (see diagram). When \(\theta = 0\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. At the instant when the string is horizontal, find the speed of \(P\) and the tension in the string.
2. At the instant when the string becomes slack, find the value of \(\theta\).

7 A particle \(P\), of mass \(m \mathrm {~kg}\), is attached to one end of a light elastic string of natural length 3.2 m and modulus of elasticity \(4 m g \mathrm {~N}\). The other end of the string is attached to a fixed point \(A\). The particle is released from rest at a point 4.8 m vertically below \(A\). At time \(t \mathrm {~s}\) after \(P\) 's release \(P\) is ( \(4 + x ) \mathrm { m }\) below \(A\).

1. Show that \(4 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 49 x\).
   \(P\) 's motion is simple harmonic.
2. Write down the amplitude of \(P\) 's motion and show that the string becomes slack instantaneously at intervals of approximately 1.8 s . A particle \(Q\) is attached to one end of a light inextensible string of length \(L \mathrm {~m}\). The other end of the string is attached to a fixed point \(B\). The particle is released from rest with the string taut and inclined at a small angle with the downward vertical. At time \(t \mathrm {~s}\) after \(Q\) 's release \(B Q\) makes an angle of \(\theta\) radians with the downward vertical.
3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } \approx - \frac { g } { L } \theta\). The period of the simple harmonic motion to which \(Q\) 's motion approximates is the same as the period of \(P\) 's motion.
4. Given that \(\theta = 0.08\) when \(t = 0\), find the speed of \(Q\) when \(t = 0.25\).