OCR M3 (Mechanics 3) 2015 June

A particle \(P\) of mass \(0.2\) kg is moving on a smooth horizontal surface with speed \(3\text{ ms}^{-1}\), when it is struck by an impulse of magnitude \(I\) Ns. The impulse acts horizontally in a direction perpendicular to the original direction of motion of \(P\), and causes the direction of motion of \(P\) to change by an angle \(\alpha\), where \(\tan \alpha = \frac{5}{12}\).

1. Show that \(I = 0.25\). [4]
2. Find the speed of \(P\) after the impulse acts. [2]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\), each of length \(2L\), are freely jointed at \(B\), and \(AB\) is freely jointed to a fixed point at \(A\). The rods are held in equilibrium in a vertical plane by a light horizontal string attached at \(C\). The rods \(AB\) and \(BC\) make angles \(\alpha\) and \(\beta\) to the horizontal respectively. The weight of rod \(BC\) is \(75\) N, and the tension in the string is \(50\) N (see diagram).

1. Show that \(\tan \beta = \frac{1}{3}\). [3]
2. Given that \(\tan \alpha = \frac{12}{5}\), find the weight of \(AB\). [5]

\includegraphics{figure_3} A small object \(P\) is attached to one end of each of two vertical light elastic strings. One string is of natural length \(0.4\) m and has modulus of elasticity \(10\) N; the other string is of natural length \(0.5\) m and has modulus of elasticity \(12\) N. The upper ends of both strings are attached to a fixed horizontal beam and \(P\) hangs in equilibrium \(0.6\) m below the beam (see diagram).

1. Show that the weight of \(P\) is \(7.4\) N and find the total elastic potential energy stored in the two strings when \(P\) is hanging in equilibrium. [6]

\(P\) is then held at a point \(0.7\) m below the beam with the strings vertical. \(P\) is released from rest.

1. Show that, throughout the subsequent motion, \(P\) performs simple harmonic motion, and find the period. [7]

A particle of mass \(0.4\) kg, moving on a smooth horizontal surface, passes through a point \(O\) with velocity \(10\text{ ms}^{-1}\). At time \(t\) s after the particle passes through \(O\), the particle has a displacement \(x\) m from \(O\), has a velocity \(v\text{ ms}^{-1}\) away from \(O\), and is acted on by a force of magnitude \(\frac{1}{5}v\) N acting towards \(O\). Find

1. the time taken for the velocity of the particle to reduce from \(10\text{ ms}^{-1}\) to \(5\text{ ms}^{-1}\), [5]
2. the average velocity of the particle over this time. [6]

\includegraphics{figure_5} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(2m\) kg and \(m\) kg respectively. The spheres are moving on a horizontal surface when they collide. Before the collision, \(A\) is moving with speed \(a\text{ ms}^{-1}\) in a direction making an angle \(\alpha\) with the line of centres and \(B\) is moving towards \(A\) with speed \(b\text{ ms}^{-1}\) in a direction making an angle \(\beta\) with the line of centres (see diagram). After the collision, \(A\) moves with velocity \(2\text{ ms}^{-1}\) in a direction perpendicular to the line of centres and \(B\) moves with velocity \(2\text{ ms}^{-1}\) in a direction making an angle of \(45°\) with the line of centres. The coefficient of restitution between \(A\) and \(B\) is \(\frac{2}{3}\).

1. Show that \(a\cos \alpha = \frac{5}{3}\sqrt{2}\) and find \(b\cos \beta\). [7]
2. Find the values of \(a\) and \(\alpha\). [4]

A particle \(P\) starts from rest from a point \(A\) and moves in a straight line with simple harmonic motion about a point \(O\). At time \(t\) seconds after the motion starts the displacement of \(P\) from \(O\) is \(x\) m towards \(A\). The particle \(P\) is next at rest when \(t = 0.25\pi\) having travelled a distance of \(1.2\) m.

1. Find the maximum velocity of \(P\). [3]
2. Find the value of \(x\) and the velocity of \(P\) when \(t = 0.7\). [4]
3. Find the other values of \(t\), for \(0 < t < 1\), at which \(P\)'s speed is the same as when \(t = 0.7\). Find also the corresponding values of \(x\). [4]

\includegraphics{figure_7} One end of a light inextensible string of length \(0.5\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.2\) kg is attached to the other end of the string. \(P\) is projected horizontally from the point \(0.5\) m below \(O\) with speed \(u\text{ ms}^{-1}\). When the string makes an angle of \(\theta\) with the downward vertical the particle has speed \(v\text{ ms}^{-1}\) (see diagram).

1. Show that, while the string is taut, the tension, \(T\) N, in the string is given by $$T = 5.88\cos \theta + 0.4u^2 - 3.92.$$ [5]
2. Find the least value of \(u\) for which the particle will move in a complete circle. [3]
3. If in fact \(u = 3.5\text{ ms}^{-1}\), find the speed of the particle at the point where the string first becomes slack. [4]

END OF QUESTION PAPER