OCR M3 (Mechanics 3) 2014 June

1 A particle \(P\) of mass 0.3 kg is moving on a smooth horizontal surface with speed \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a horizontal impulse. The magnitude of the impulse is 0.6 Ns .

1. (a) Find the greatest possible speed of \(P\) after the impulse acts.
   (b) Find the least possible speed of \(P\) after the impulse acts.
2. In fact the speed of \(P\) after the impulse acts is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the angle the impulse makes with the original direction of travel of \(P\) and draw a sketch to make this direction clear.

2 One end of a light elastic string, of natural length 0.6 m and modulus of elasticity 30 N , is attached to a fixed point \(O\). A particle \(P\) of weight 48 N is attached to the other end of the string. \(P\) is released from rest at a point \(d \mathrm {~m}\) vertically below \(O\). Subsequently \(P\) just reaches \(O\).

1. Find \(d\).
2. Find the magnitude and direction of the acceleration of \(P\) when it has travelled 1.3 m from its point of release.

3
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-2_403_951_1247_559} 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 0.4 kg . Immediately before the collision \(A\) is moving with speed \(2.8 \mathrm {~ms} ^ { - 1 }\) along the line of centres, and \(B\) is moving with speed \(1 \mathrm {~ms} ^ { - 1 }\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision \(A\) is stationary. Find

1. the coefficient of restitution between \(A\) and \(B\),
2. the angle turned through by the direction of motion of \(B\) as a result of the collision. \section*{\(\mathrm { OCR } ^ { \text {勾 } }\)}

4 A particle \(P\) of mass 0.4 kg is projected horizontally with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface. At time \(t \mathrm {~s}\) after projection \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving away from \(O\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force of magnitude \(1.6 v ^ { 2 } \mathrm {~N}\) resisting the motion of \(P\).

1. Find an expression for \(\frac { \mathrm { d } v } { \mathrm {~d} x }\) in terms of \(v\), and hence show that \(v = 2 \mathrm { e } ^ { - 4 x }\).
2. Find the distance travelled by \(P\) in the 0.5 seconds after it leaves \(O\).

5
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-3_510_716_662_676} Two uniform rods \(A B\) and \(B C\), each of length \(4 L\), are freely jointed at \(B\), and rest in a vertical plane with \(A\) and \(C\) on a smooth horizontal surface. The weight of \(A B\) is \(W\) and the weight of \(B C\) is \(2 W\). The rods are joined by a horizontal light inextensible string fixed to each rod at a point distance \(L\) from \(B\), so that each rod is inclined at an angle of \(60 ^ { \circ }\) to the horizontal (see diagram).

1. By considering the equilibrium of the whole body, show that the force acting on \(B C\) at \(C\) is \(1.75 W\) and find the force acting on \(A B\) at \(A\).
2. Find the tension in the string in terms of \(W\).
3. Find the horizontal and vertical components of the force acting on \(A B\) at \(B\), and state the direction of the component in each case.

6
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_547_515_267_772} A hollow cylinder is fixed with its axis horizontal. \(O\) is the centre of a vertical cross-section of the cylinder and \(D\) is the highest point on the cross-section. \(A\) and \(C\) are points on the circumference of the cross-section such that \(A O\) and \(C O\) are both inclined at an angle of \(30 ^ { \circ }\) below the horizontal diameter through \(O\). The inner surface of the cylinder is smooth and has radius 0.8 m (see diagram). A particle \(P\), of mass \(m \mathrm {~kg}\), and a particle \(Q\), of mass \(5 m \mathrm {~kg}\), are simultaneously released from rest from \(A\) and \(C\), respectively, inside the cylinder. \(P\) and \(Q\) collide; the coefficient of restitution between them is 0.95 .

1. Show that, immediately after the collision, \(P\) moves with speed \(6.3 \mathrm {~ms} ^ { - 1 }\), and find the speed and direction of motion of \(Q\).
2. Find, in terms of \(m\), an expression for the normal reaction acting on \(P\) when it subsequently passes through \(D\).

7
\includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-4_382_773_1567_648} One end of a light elastic string, of natural length 0.3 m , is attached to a fixed point \(O\) on a smooth plane that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.2\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string lies along a line of greatest slope of the plane and has modulus of elasticity \(2.45 m \mathrm {~N}\) (see diagram).

1. Show that in the equilibrium position the extension of the string is 0.24 m .
   \(P\) is given a velocity of \(0.3 \mathrm {~ms} ^ { - 1 }\) down the plane from the equilibrium position.
2. Show that \(P\) performs simple harmonic motion with period 2.20 s (correct to 3 significant figures), and find the amplitude of the motion.
3. Find the distance of \(P\) from \(O\) and the velocity of \(P\) at the instant 1.5 seconds after \(P\) is set in motion.