\(P\) is a passenger on a roundabout at a fair. When the roundabout is rotating uniformly, a given point \(A\) on the roundabout moves in a circle of radius \(3a\) about the central axis of the roundabout with constant angular velocity \(\omega\). The passenger \(P\) is moving relative to the roundabout in a circle of radius \(a\) about \(A\) with constant angular velocity \(2\omega\). How many times is \(P\) stationary during one revolution of \(A\)? Find the distance travelled by \(P\) between two points when he is at rest.
A square-wheeled bicycle is ridden at constant horizontal speed \(V\). The sides of one wheel are always parallel to the sides of the other, and the wheels do not slip on the ground. If the wheels remain in contact with the ground show that \[V^2 < ag/16,\] where \(a\) is the circumference of each wheel.
Identical ball-bearings \(A\), \(B\), \(C\), of diameter \(a\), are collinear. \(B\) and \(C\) are initially at rest with their centres a distance \(b\) apart, and \(A\), moving co-linearly towards them, strikes \(B\). The coefficient of restitution is \(e\). Show that \(A\) will strike \(B\) again, after \(A\) has travelled a further distance \[\frac{1+e}{1-e}(b-a).\]
Suppose that the coefficient of friction between two surfaces is directly proportional to the velocity difference between the surfaces (the 'slipping velocity'). Show that a body can slide down an inclined plane with a constant velocity which depends on the inclination of the plane. A cylinder of radius \(a\) and radius of gyration \(k\), with its axis horizontal, rolls (with slipping of the above character) down an inclined plane. If the cylinder starts from rest, determine the subsequent motion. Show that the frictional force tends to the force which would be necessary to keep the cylinder rolling without slipping, but that the slipping velocity at the point of contact does not tend to zero.
A particle of unit mass moves, in the absence of gravity, in the plane of a disc of unit radius and moment of inertia \(k^2\). The particle is attached by a light inextensible string of length \(l_0\) to a point on the rim of the disc. The particle's motion is such that the string is always taut, and wraps itself round the disc as the particle moves in a clockwise sense. (i) If the disc is fixed, show that the kinetic energy \(T\) of the system is \(\frac{1}{2}l^2\dot{\theta}^2\), and its angular momentum \(h\) is \(l^2\dot{\theta}\), where \(l\) is the length of the unwrapped portion of the string. Which, if either, of \(T\) and \(h\) is constant? (ii) If the disc is free to rotate about its axis, with angular velocity \(\dot{\phi}\) which is positive if the body rotates anticlockwise, show that \[h = l^2\dot{\theta}+(1+k^2+l^2)\dot{\phi},\] and find \(T\). Which, if either, of \(T\) and \(h\) is now constant? Show that \[h^2+l^2\dot{\theta}^2(1+k^2) = 2T(1+k^2+l^2).\]
Show that \(|{\bf a} \wedge {\bf b}|^2 = a^2b^2 - ({\bf a} \cdot {\bf b})^2\). If \({\bf a} \wedge {\bf b} \neq 0\), and if \[\alpha{\bf a} + \beta{\bf b} + \gamma{\bf a} \wedge {\bf b} = \alpha'{\bf a} + \beta'{\bf b} + \gamma'{\bf a} \wedge {\bf b},\] show that \(\alpha = \alpha'\), \(\beta = \beta'\), \(\gamma = \gamma'\). For some \(\lambda\), and for some non-zero \({\bf x}\), \[({\bf a}\cdot{\bf x}){\bf a}+({\bf b}\cdot{\bf x}){\bf b} = \lambda{\bf x}.\] By looking for solutions of the form \({\bf x} = \alpha{\bf a} + \beta{\bf b} + \gamma{\bf a} \wedge {\bf b}\), or otherwise, show that either \(\lambda = 0\) or \[\lambda^2-(a^2+b^2)\lambda + |{\bf a} \wedge {\bf b}|^2 = 0.\]