OCR M3 (Mechanics 3) 2010 January

1
\includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_323_639_255_753} A particle \(P\) of mass 0.4 kg is moving horizontally with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it receives an impulse of magnitude \(I \mathrm { Ns }\), in a direction which makes an angle \(( 180 - \theta ) ^ { \circ }\) with the direction of motion of \(P\). Immediately after the impulse acts \(P\) moves horizontally with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The direction of motion of \(P\) is turned through an angle of \(60 ^ { \circ }\) by the impulse (see diagram). Find \(I\) and \(\theta\).

2
\includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-2_421_759_936_694} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 2 kg and 3 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving along the line of centres, and \(B\) has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres (see diagram). The coefficient of restitution is 0.6 . The direction of motion of \(B\) after the collision makes an angle of \(45 ^ { \circ }\) with the line of centres. Find the value of \(v\).

3
\includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-3_812_773_260_685} Two uniform rods \(A B\) and \(B C\), each of length \(2 a\), have weights \(2 W\) and \(W\) respectively. The rods are freely jointed to each other at \(B\), and \(B C\) is freely jointed to a fixed point at \(C\). The rods are held in equilibrium in a vertical plane by a light string attached to \(A\) and perpendicular to \(A B\). The rods \(A B\) and \(B C\) make angles \(45 ^ { \circ }\) and \(\alpha\), respectively, with the horizontal. The tension in the string is \(T\) (see diagram).

1. By taking moments about \(B\) for \(A B\), show that \(W = \sqrt { 2 } T\).
2. Find the value of \(\tan \alpha\).

4 A particle \(P\) of mass 0.2 kg travels in a straight line on a horizontal surface. It passes through a point \(O\) on the surface with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A resistive force of magnitude \(0.2 \left( v + v ^ { 2 } \right) \mathrm { N }\) acts on \(P\) in the direction opposite to its motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) when it is at a distance \(x \mathrm {~m}\) from \(O\).

1. Show that \(\frac { 1 } { 1 + v } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 1\).
2. By solving the differential equation in part (i) show that \(\frac { - \mathrm { e } ^ { x } } { 3 - \mathrm { e } ^ { x } } \frac { \mathrm {~d} x } { \mathrm {~d} t } = - 1\), where \(t\) s is the time taken for \(P\) to travel \(x \mathrm {~m}\) from \(O\).
3. Hence find the value of \(t\) when \(x = 1\).

5 A light elastic string of natural length 1.6 m has modulus of elasticity 120 N . One end of the string is attached to a fixed point \(O\) and the other end is attached to a particle \(P\) of weight 1.5 N . The particle is released from rest at the point \(A\), which is 2.1 m vertically below \(O\). It comes instantaneously to rest at \(B\), which is vertically above \(O\).

1. Verify that the distance \(A B\) is 4 m .
2. Find the maximum speed of \(P\) during its upward motion from \(A\) to \(B\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_351_442_303_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08760a55-da6c-41f2-a88a-289ecc227f69-4_394_648_260_1018} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A light inextensible string of length \(0.8 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.4 kg and 0.58 kg respectively, attached to its ends. The string passes over a smooth horizontal cylinder of radius 0.8 m , which is fixed with its axis horizontal and passing through a fixed point \(O\). The string is held at rest in a vertical plane perpendicular to the axis of the cylinder, with \(P\) and \(Q\) at opposite ends of the horizontal diameter of the cylinder through \(O\) (see Fig. 1). The string is released and \(Q\) begins to descend. When \(O P\) has rotated through \(\theta\) radians, with \(P\) remaining in contact with the cylinder, the speed of each particle is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).

1. By considering the total energy of the system, obtain an expression for \(v ^ { 2 }\) in terms of \(\theta\).
2. Show that the magnitude of the force exerted on \(P\) by the cylinder is \(( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }\).
3. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(1.53 < \alpha < 1.54\).

7 A particle \(P\) of mass 0.5 kg is attached to one end of each of two identical light elastic strings of natural length 1.6 m and modulus of elasticity 19.6 N . The other ends of the strings are attached to fixed points \(A\) and \(B\) on a line of greatest slope of a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The distance \(A B\) is 4.8 m and \(A\) is higher than \(B\).

1. Find the distance \(A P\) for which \(P\) is in equilibrium on the line \(A B\).
   \(P\) is released from rest at a point on \(A B\) where both strings are taut. The strings remain taut during the subsequent motion of \(P\) and \(t\) seconds after release the distance \(A P\) is \(( 2.5 + x ) \mathrm { m }\).
2. Use Newton's second law to obtain an equation of the form \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = k x\). State the property of the constant \(k\) for which the equation indicates that \(P\) 's motion is simple harmonic, and find the period of this motion.
3. Given that \(x = 0.5\) when \(t = 0\), find the values of \(x\) for which the speed of \(P\) is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). {www.ocr.org.uk}) after the live examination series.
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