OCR M3 (Mechanics 3) 2013 June

1
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-2_435_665_255_699} A small object \(W\) of weight 100 N is attached to one end of each of two parallel light elastic strings. One string is of natural length 0.4 m and has modulus of elasticity 20 N ; the other string is of natural length 0.6 m and has modulus of elasticity 30 N . The upper ends of both strings are attached to a horizontal ceiling and \(W\) hangs in equilibrium at a distance \(d \mathrm {~m}\) below the ceiling (see diagram). Find \(d\).

2 A particle of mass 0.3 kg is projected horizontally under gravity with velocity \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point 0.4 m above a smooth horizontal plane. The particle first hits the plane at point \(A\); it bounces and hits the plane a second time at point \(B\). The distance \(A B\) is 1 m . Calculate

1. the vertical component of the velocity of the particle when it arrives at \(A\), and the time taken for the particle to travel from \(A\) to \(B\),
2. the coefficient of restitution between the particle and the plane,
3. the impulse exerted by the plane on the particle at \(A\).

3 A particle \(P\) of mass 0.2 kg moves on a smooth horizontal plane. Initially it is projected with velocity \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) towards another fixed point \(A\). At time \(t\) s after projection, \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), with the direction \(O A\) being positive. A force of \(( 1.5 t - 1 ) \mathrm { N }\) acts on \(P\) in the direction parallel to \(O A\).

1. Find an expression for \(v\) in terms of \(t\).
2. Find the time when the velocity of \(P\) is next \(0.8 \mathrm {~ms} ^ { - 1 }\).
3. Find the times when \(P\) subsequently passes through \(O\).
4. Find the distance \(P\) travels in the third second of its motion.

4 Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.1 kg and \(B\) has mass 0.2 kg . Immediately before the collision \(A\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving away from \(A\) with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, where \(\cos \theta = 0.6\) (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_422_844_431_612} The coefficient of restitution between the spheres is 0.8 . Find

1. the velocity of \(A\) immediately after the collision,
2. the angle turned through by the direction of motion of \(B\) as a result of the collision.

5
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-3_449_442_1281_794} A fixed smooth sphere of radius 0.6 m has centre \(O\) and highest point \(T\). A particle of mass \(m \mathrm {~kg}\) is released from rest at a point \(A\) on the sphere, such that angle \(T O A\) is \(\frac { \pi } { 6 }\) radians. The particle leaves the surface of the sphere at \(B\) (see diagram).

1. Show that \(\cos T O B = \frac { \sqrt { 3 } } { 3 }\).
2. Find the speed of the particle at \(B\).
3. Find the transverse acceleration of the particle at \(B\).

6 Two uniform rods \(A B\) and \(B C\), each of length \(2 l\), are freely jointed at \(B\). The weight of \(A B\) is \(W\) and the weight of \(B C\) is \(2 W\). The rods are in a vertical plane with \(A\) freely pivoted at a fixed point and \(C\) resting in equilibrium on a rough horizontal plane. The normal and frictional components of the force acting on \(B C\) at \(C\) are \(R\) and \(F\) respectively. The rod \(A B\) makes an angle \(30 ^ { \circ }\) to the horizontal and the rod \(B C\) makes an angle \(60 ^ { \circ }\) to the horizontal (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-4_682_901_479_587}

1. By considering the equilibrium of \(\operatorname { rod } B C\), show that \(W + \sqrt { 3 } F = R\).
2. By taking moments about \(A\) for the equilibrium of the whole system, find another equation involving \(W , F\) and \(R\).
3. Given that the friction at \(C\) is limiting, calculate the value of the coefficient of friction at \(C\).

7 A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 39.2 mN . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).

1. Show that, while the string is in tension, the particle performs simple harmonic motion about a point 1 m below \(O\).
2. Show that when \(P\) is at its lowest point the extension of the string is 0.8 m .
3. Find the time after its release that \(P\) first reaches its lowest point.
4. Find the velocity of \(P 0.8 \mathrm {~s}\) after it is released from \(O\). OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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