OCR M3 (Mechanics 3) 2007 January

1 A particle \(P\) of mass 0.6 kg is attached to a fixed point \(O\) by a light inextensible string of length 0.4 m . While hanging at a distance 0.4 m vertically below \(O , P\) is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in a complete vertical circle. Calculate the tension in the string when \(P\) is vertically above \(O\).

2
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_231_971_539_587} When a tennis ball of mass 0.057 kg bounces it receives an impulse of magnitude \(I \mathrm {~N} \mathrm {~s}\) at an angle of \(\theta\) to the horizontal. Immediately before the ball bounces it has speed \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal. Immediately after the ball bounces it has speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction of \(30 ^ { \circ }\) to the horizontal (see diagram). Find \(I\) and \(\theta\).

3
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_465_757_1146_694} Two identical uniform rods, \(A B\) and \(B C\), are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in limiting equilibrium in a vertical plane, with \(C\) resting on a rough horizontal surface. \(A B\) is horizontal, and \(B C\) is inclined at \(60 ^ { \circ }\) to the horizontal. The weight of each rod is 160 N (see diagram).

1. By taking moments for \(A B\) about \(A\), find the vertical component of the force on \(A B\) at \(B\). Hence or otherwise find the magnitude of the vertical component of the contact force on \(B C\) at \(C\). [3]
2. Calculate the magnitude of the frictional force on \(B C\) at \(C\) and state its direction.
3. Calculate the value of the coefficient of friction at \(C\).

4 A particle \(P\) of mass 0.2 kg is suspended from a fixed point \(O\) by a light elastic string of natural length 0.7 m and modulus of elasticity \(3.5 \mathrm {~N} . P\) is at the equilibrium position when it is projected vertically downwards with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after being set in motion \(P\) is \(x \mathrm {~m}\) below the equilibrium position and has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the equilibrium position of \(P\) is 1.092 m below \(O\).
2. Prove that \(P\) moves with simple harmonic motion, and calculate the amplitude.
3. Calculate \(x\) and \(v\) when \(t = 0.4\).

5 The pilot of a hot air balloon keeps it at a fixed altitude by dropping sand from the balloon. Each grain of sand has mass \(m \mathrm {~kg}\) and is released from rest. When a grain has fallen a distance \(x \mathrm {~m}\), it has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Each grain falls vertically and the only forces acting on it are its weight and air resistance of magnitude \(m k v ^ { 2 } \mathrm {~N}\), where \(k\) is a positive constant.

1. Show that \(\left( \frac { v } { g - k v ^ { 2 } } \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\).
2. Find \(v ^ { 2 }\) in terms of \(k , g\) and \(x\). Hence show that, as \(x\) becomes large, the limiting value of \(v\) is \(\sqrt { \frac { g } { k } }\).
3. Given that the altitude of the balloon is 300 m and that each grain strikes the ground at \(90 \%\) of its limiting velocity, find \(k\).

6
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-3_446_821_1007_664} Two uniform smooth spheres \(A\) and \(B\) of equal radius are moving on a horizontal surface when they collide. \(A\) has mass 0.4 kg , and \(B\) has mass \(m \mathrm {~kg}\). Immediately before the collision, \(A\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an acute angle \(\theta\) to the line of centres, and \(B\) is moving with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(30 ^ { \circ }\) to the line of centres. Immediately after the collision \(A\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(45 ^ { \circ }\) to the line of centres, and \(B\) is moving with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the line of centres (see diagram).

1. Find \(u\).
2. Given that \(\theta = 88.1 ^ { \circ }\) correct to 1 decimal place, calculate the approximate values of \(v\) and \(m\).
3. The coefficient of restitution is 0.75 . Show that the exact value of \(\theta\) is a root of the equation \(8 \sin \theta - 6 \cos \theta = 9 \cos 30 ^ { \circ }\).

7
\includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-4_721_691_269_726} The diagram shows a particle \(P\) of mass 0.5 kg attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius 1.2 m . The string has natural length 0.6 m and modulus of elasticity \(6.86 \mathrm {~N} . P\) is released from rest at a point on the surface of the sphere where the acute angle \(A O P\) is at least 0.5 radians.

1. (a) For the case angle \(A O P = \alpha , P\) remains at rest. Show that \(\sin \alpha = 2.8 \alpha - 1.4\).
   (b) Use the iterative formula $$\alpha _ { n + 1 } = \frac { \sin \alpha _ { n } } { 2.8 } + 0.5 ,$$ with \(\alpha _ { 1 } = 0.8\), to find \(\alpha\) correct to 2 significant figures.
2. Given instead that angle \(A O P = 0.5\) radians when \(P\) is released, find the speed of \(P\) when angle \(A O P = 0.8\) radians, given that \(P\) is at all times in contact with the surface of the sphere. State whether the speed of \(P\) is increasing or decreasing when angle \(A O P = 0.8\) radians.