OCR M3 (Mechanics 3) 2012 June

2
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-3_442_636_255_715}
\(B\) is a point on a smooth plane surface inclined at an angle of \(15 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.45 kg is released from rest at the point \(A\) which is 2.5 m vertically above \(B\). The particle \(P\) rebounds from the surface at an angle of \(60 ^ { \circ }\) to the line of greatest slope through \(B\), with a speed of \(u \mathrm {~ms} ^ { - 1 }\). The impulse exerted on \(P\) by the surface has magnitude \(I\) Ns and is in a direction making an angle of \(\theta ^ { \circ }\) with the upward vertical through \(B\) (see diagram).

1. Explain why \(\theta = 15\).
2. Find the values of \(u\) and \(I\).

3 A particle \(P\) of mass \(m \mathrm {~kg}\) is released from rest and falls vertically. When \(P\) has fallen a distance of \(x \mathrm {~m}\) it has a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The only forces acting on \(P\) are its weight and air resistance of magnitude \(\frac { 1 } { 400 } m v ^ { 2 } \mathrm {~N}\).

1. Find \(v ^ { 2 }\) in terms of \(x\) and show that \(v ^ { 2 }\) must be less than 3920 .
2. Find the speed of \(P\) when it has fallen 100 m .

4
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-4_524_611_255_703} A hollow cylinder is fixed with its axis horizontal. The inner surface of the cylinder is smooth and has radius 0.6 m . A particle \(P\) of mass 0.45 kg is projected horizontally with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the lowest point of a vertical cross-section of the cylinder and moves in the plane of the cross-section, which is perpendicular to the axis of the cylinder. While \(P\) remains in contact with the surface, its speed is \(v \mathrm {~ms} ^ { - 1 }\) when \(O P\) makes an angle \(\theta\) with the downward vertical at \(O\), where \(O\) is the centre of the cross-section (see diagram). The force exerted on \(P\) by the surface is \(R \mathrm {~N}\).

1. Show that \(v ^ { 2 } = 4.24 + 11.76 \cos \theta\) and find an expression for \(R\) in terms of \(\theta\).
2. Find the speed of \(P\) at the instant when it leaves the surface.

5 One end of a light elastic string, of natural length 0.78 m and modulus of elasticity 0.8 mgN , is attached to a fixed point \(O\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. \(P\) is released from rest at \(O\) and moves down the plane without reaching the bottom. Find

1. the maximum speed of \(P\) in the subsequent motion,
2. the distance of \(P\) from \(O\) when it is at its lowest point.

6
\includegraphics[max width=\textwidth, alt={}, center]{cc74a925-f1c8-4f59-a421-b46444cae5ec-5_387_867_258_575} Two smooth uniform spheres \(A\) and \(B\), of equal radius, have masses 2 kg and \(m \mathrm {~kg}\) respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving towards \(B\) at an angle of \(\alpha\) to the line of centres, where \(\cos \alpha = 0.6\). \(B\) has speed \(2 \mathrm {~ms} ^ { - 1 }\) and is moving towards \(A\) along the line of centres (see diagram). As a result of the collision, \(A\) 's loss of kinetic energy is \(7.56 \mathrm {~J} , B\) 's direction of motion is reversed and \(B\) 's speed after the collision is \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the speed of \(A\) after the collision,
2. the component of \(A\) 's velocity after the collision, parallel to the line of centres, stating with a reason whether its direction is to the left or to the right,
3. the value of \(m\),
4. the coefficient of restitution between \(A\) and \(B\).
   \(7 S _ { A }\) and \(S _ { B }\) are light elastic strings. \(S _ { A }\) has natural length 2 m and modulus of elasticity \(120 \mathrm {~N} ; S _ { B }\) has natural length 3 m and modulus of elasticity 180 N . A particle \(P\) of mass 0.8 kg is attached to one end of each of the strings. The other ends of \(S _ { A }\) and \(S _ { B }\) are attached to fixed points \(A\) and \(B\) respectively, on a smooth horizontal table. The distance \(A B\) is \(6 \mathrm {~m} . P\) is released from rest at the point of the line segment \(A B\) which is 2.9 m from \(A\).
5. For the subsequent motion, show that the total elastic potential energy of the strings is the same when \(A P = 2.1 \mathrm {~m}\) and when \(A P = 2.9 \mathrm {~m}\). Deduce that neither string becomes slack.
6. Find, in terms of \(x\), an expression for the acceleration of \(P\) in the direction of \(A B\) when \(A P = ( 2.5 + x ) \mathrm { m }\).
7. State, giving a reason, the type of motion of \(P\) and find the time taken between successive occasions when \(P\) is instantaneously at rest. For the instant 0.6 seconds after \(P\) is released, find
8. the distance travelled by \(P\),
9. the speed of \(P\).