OCR Further Mechanics (Further Mechanics) 2018 September

1 A car of mass 850 kg is being driven uphill along a straight road inclined at \(7 ^ { \circ }\) to the horizontal. The resistance to motion is modelled as a constant force of magnitude 140 N . At a certain instant the car's speed is \(12 \mathrm {~ms} ^ { - 1 }\) and its acceleration is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Calculate the power of the car's engine at this instant.
2. Find the constant speed at which the car could travel up the hill with the engine generating this power.

2 A particle of mass 0.8 kg is moving in a straight line on a smooth horizontal surface with constant speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a horizontal impulse. Immediately after the impulse acts, the particle is moving with speed \(9 \mathrm {~ms} ^ { - 1 }\) at an angle of \(50 ^ { \circ }\) to its original direction of motion (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-2_293_597_989_735} Find

1. the magnitude of the impulse,
2. the angle that the impulse makes with the original direction of motion of the particle.

3 Assume that the earth moves round the sun in a circle of radius \(1.50 \times 10 ^ { 8 } \mathrm {~km}\) at constant speed, with one complete orbit taking 365 days. Given that the mass of the earth is \(5.97 \times 10 ^ { 24 } \mathrm {~kg}\),

1. calculate the magnitude of the force exerted by the sun on the earth, giving your answer in newtons,
2. state the direction in which this force acts.
   \(4 \quad A\) and \(B\) are two points a distance of 5 m apart on a horizontal ceiling. A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to \(A\) and \(B\) by light elastic strings. The particle hangs in equilibrium at a distance of 4 m from \(A\) and 3 m from \(B\) so that angle \(A P B = 90 ^ { \circ }\) (see diagram).
   \includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-3_286_745_402_660} The string joining \(P\) to \(A\) has natural length 2 m and modulus of elasticity \(\lambda _ { A } \mathrm {~N}\). The string joining \(P\) to \(B\) also has natural length 2 m but has modulus of elasticity \(\lambda _ { B } \mathrm {~N}\).
3. (a) Show that \(\lambda _ { B } = \frac { 8 } { 3 } \lambda _ { A }\).
   (b) Find an expression for \(\lambda _ { A }\) in terms of \(m\) and \(g\).
4. Find, in terms of \(m\) and \(g\), the total elastic potential energy stored in the strings. The string joining \(P\) to \(A\) is detached from \(A\) and a second particle, \(Q\), of mass \(0.3 m \mathrm {~kg}\) is attached to the free end of the string. \(Q\) is then gently lowered into a position where the system hangs vertically in equilibrium.
5. Find the distance of \(Q\) below \(B\) in this equilibrium position.

5 One end of a non-uniform rod is freely hinged to a fixed point so that the rod can rotate about the point. When the rod rotates with angular velocity \(\omega\) it can be shown that the kinetic energy \(E\) of the rod is given by \(E = \frac { 1 } { 2 } I \omega ^ { 2 }\), where \(I\) is a quantity called the moment of inertia of the rod.

1. Deduce the dimensions of \(I\).
2. Given that the rod has mass \(m\) and length \(r\), suggest an expression for \(I\), explaining any additional symbols that you use. A student notices that the formula \(E = \frac { 1 } { 2 } I \omega ^ { 2 }\) looks similar to the formula \(E = \frac { 1 } { 2 } m v ^ { 2 }\) for the kinetic energy of a particle, with angular velocity for the rod corresponding to velocity for the particle, and moment of inertia corresponding to mass. Assuming a similar correspondence between angular acceleration (i.e. \(\frac { \mathrm { d } \omega } { \mathrm { d } t }\) ) and acceleration, the student thinks that an equation for angular motion of the rod corresponding to Newton's second law for the particle should be \(F = I \alpha\), where \(F\) is the force applied to the rod and \(\alpha\) is the resulting angular acceleration.
3. Use dimensional analysis to show that the student's suggestion is incorrect.
4. State the dimensions of a quantity \(x\) for which the equation \(F x = I \alpha\) would be dimensionally consistent.
5. Explain why the fact that the equation in part (iv) is dimensionally consistent does not necessarily mean that it is correct.

6 A particle \(P\) of mass \(m\) moves along the positive \(x\)-axis. When its displacement from the origin \(O\) is \(x\) its velocity is \(v\), where \(v \geqslant 0\). It is subject to two forces: a constant force \(T\) in the positive \(x\) direction, and a resistive force which is proportional to \(v ^ { 2 }\).

1. Show that \(v ^ { 2 } = \frac { 1 } { k } \left( T - A \mathrm { e } ^ { - \frac { 2 k x } { m } } \right)\) where \(A\) and \(k\) are constants.
   \(P\) starts from rest at \(O\).
2. Find an expression for the work done against the resistance to motion as \(P\) moves from \(O\) to the point where \(x = 1\).
3. Find an expression for the limiting value of the velocity of \(P\) as \(x\) increases.

7 A uniform solid hemisphere has radius 0.4 m . A uniform solid cone, made of the same material, has base radius 0.4 m and height 1.2 m . A solid, \(S\), is formed by joining the hemisphere and the cone so that their circular faces coincide. \(O\) is the centre of the joint circular face and \(V\) is the vertex of the cone. \(G\) is the centre of mass of \(S\).

1. Explain briefly why \(G\) lies on the line through \(O\) and \(V\).
2. Show that the distance of \(G\) from \(O\) is 0.12 m .
   (The volumes of a hemisphere and cone are \(\frac { 2 } { 3 } \pi r ^ { 3 }\) and \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) respectively.)
   \includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-4_474_719_1347_664}
   \(S\) is suspended from two light vertical strings, one attached to \(V\) and the other attached to a point on the circumference of the joint circular face, and hangs in equilibrium with OV horizontal (see diagram).
3. The weight of \(S\) is \(W\). Find the magnitudes of the tensions in the strings in terms of \(W\).

8 A point \(O\) is situated a distance \(h\) above a smooth horizontal plane, and a particle \(A\) of mass \(m\) is attached to \(O\) by a light inextensible string of length \(h\). A particle \(B\) of mass \(2 m\) is at rest on the plane, directly below \(O\), and is attached to a point \(C\) on the plane, where \(B C = l\), by a light inextensible string of length \(l . A\) is released from rest with the string \(O A\) taut and making an acute angle \(\theta\) with the downward vertical (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{19c3a9d0-15b6-4dd0-a00b-577c3fd2cf52-5_604_1137_486_552}
\(A\) moves in a vertical plane perpendicular to \(C B\) and collides directly with \(B\). As a result of this collision, \(A\) is brought to rest and \(B\) moves on the plane in a horizontal circle with centre \(C\). After \(B\) has made one complete revolution the particles collide again.

1. Show that, on the next occasion that \(A\) comes to rest, the string \(O A\) makes an angle \(\phi\) with the downward vertical through \(O\), where \(\cos \phi = \frac { 3 + \cos \theta } { 4 }\).
   \(A\) and \(B\) collide again when \(A O\) is next vertical.
2. Find the percentage of the original energy of the system that remains immediately after this collision.
3. Explain why the total momentum of the particles immediately before the first collision is the same as the total momentum of the particles immediately after the second collision.
4. Explain why the total momentum of the particles immediately before the first collision is different from the total momentum of the particles immediately after the third collision. \section*{OCR} \section*{Oxford Cambridge and RSA}