OCR M3 (Mechanics 3) 2011 June

\includegraphics{figure_1} A particle \(P\) of mass \(0.3\) kg is moving in a straight line with speed \(4\) m s\(^{-1}\) when it is deflected through an angle \(\theta\) by an impulse of magnitude \(I\) N s. The impulse acts at right angles to the initial direction of motion of \(P\) (see diagram). The speed of \(P\) immediately after the impulse acts is \(5\) m s\(^{-1}\). Show that \(\cos \theta = 0.8\) and find the value of \(I\). [4]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), of lengths \(3\) m and \(4\) m respectively, have weights \(300\) N and \(400\) N respectively. The rods are freely jointed at \(A\). The mid-points of the rods are joined by a light inextensible string. The rods are in equilibrium in a vertical plane with the string taut and \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is \(2.4\) m above the surface (see diagram).

1. Show that the force exerted by the surface on \(AB\) is \(374\) N and find the force exerted by the surface on \(AC\). [4]
2. Find the tension in the string. [3]
3. Find the horizontal and vertical components of the force exerted on \(AB\) at \(A\) and state their directions. [3]

A particle \(P\) of mass \(0.25\) kg is projected horizontally with speed \(5\) m s\(^{-1}\) from a fixed point \(O\) on a smooth horizontal surface and moves in a straight line on the surface. The only horizontal force acting on \(P\) has magnitude \(0.2v^2\) N, where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s after it is projected from \(O\). This force is directed towards \(O\).

1. Find an expression for \(v\) in terms of \(t\). [5]

The particle \(P\) passes through a point \(X\) with speed \(0.2\) m s\(^{-1}\).

1. Find the average speed of \(P\) for its motion between \(O\) and \(X\). [5]

One end of a light inextensible string of length \(2\) 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 held at rest with the string taut so that \(OP\) makes an angle of \(0.15\) radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(OP\) makes an angle of \(\theta\) radians with the downward vertical.

1. Show that \(\frac{d^2\theta}{dt^2} = -4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic. [3]

Using the simple harmonic approximation,

1. obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = -0.1\), [5]
2. find the angular speed of \(OP\) and the linear speed of \(P\) when \(t = 0.5\). [3]

\includegraphics{figure_5} Two uniform smooth identical spheres \(A\) and \(B\) are moving towards each other on a horizontal surface when they collide. Immediately before the collision \(A\) and \(B\) are moving with speeds \(u_A\) m s\(^{-1}\) and \(u_B\) m s\(^{-1}\) respectively, at acute angles \(\alpha\) and \(\beta\), respectively, to the line of centres. Immediately after the collision \(A\) and \(B\) are moving with speeds \(v_A\) m s\(^{-1}\) and \(v_B\) m s\(^{-1}\) respectively, at right angles and at acute angle \(\gamma\), respectively, to the line of centres (see diagram).

1. Given that \(\sin \beta = 0.96\) and \(\frac{v_B}{u_B} = 1.2\), find the value of \(\sin \gamma\). [2]
2. Given also that, before the collision, the component of \(A\)'s velocity parallel to the line of centres is \(2\) m s\(^{-1}\), find the values of \(u_B\) and \(v_B\). [5]
3. Find the coefficient of restitution between the spheres. [3]
4. Given that the kinetic energy of \(A\) immediately before the collision is \(6.5m\) J, where \(m\) kg is the mass of \(A\), find the value of \(v_A\). [2]

\includegraphics{figure_6} A particle \(P\) of weight \(6\) N is attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius \(0.8\) m. The string has natural length \(\frac{1}{10}\pi\) m and modulus of elasticity \(9\) N. \(P\) is released from rest at a point \(X\) on the sphere where \(OX\) makes an angle of \(\frac{1}{3}\pi\) radians with the upwards vertical. \(P\) remains in contact with the sphere as it moves upwards to \(A\). At time \(t\) seconds after the release, \(OP\) makes an angle of \(\theta\) radians with the upwards vertical (see diagram). When \(\theta = \frac{1}{4}\pi\), \(P\) passes through the point \(Y\).

1. Show that as \(P\) moves from \(X\) to \(Y\) its gravitational potential energy increases by \(2.4(\sqrt{3} - \sqrt{2})\) J and the elastic potential energy in the string decreases by \(0.4\pi\) J. [5]
2. Verify that the transverse acceleration of \(P\) is zero when \(\theta = \frac{1}{4}\pi\), and hence find the maximum speed of \(P\). [6]

One end of a light inextensible string of length \(0.8\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.5\) kg is attached to the other end of the string. \(P\) is projected horizontally from the point \(0.8\) m vertically below \(O\) with speed \(5.6\) m s\(^{-1}\). \(P\) starts to move in a vertical circle with centre \(O\). The speed of \(P\) is \(v\) m s\(^{-1}\) when the string makes an angle \(\theta\) with the downward vertical.

1. While the string remains taut, show that \(v^2 = 15.68(1 + \cos \theta)\), and find the tension in the string in terms of \(\theta\). [7]
2. For the instant when the string becomes slack, find the value of \(\theta\) and the value of \(v\). [3]
3. Find, in either order, the speed of \(P\) when it is at its greatest height after the string becomes slack, and the greatest height reached by \(P\) above its point of projection. [4]