A railway truck of total mass \(M\) has identical wheels of radius \(a\) whose combined moment of inertia, about the axles is \(J\). The axle bearings are frictionless, but the coefficient of limiting friction between the wheels and the rails is \(\mu\). The truck is on a horizontal track, and is pulled by a force \(P\). The vertical acceleration due to gravity is \(g\).
A weightless rod carries a particle of mass \(m\) at its upper end. It is balanced in unstable equilibrium on a rough horizontal table, and begins to fall sideways. Using conservation of energy, find the angular velocity (squared) and the angular acceleration as functions of the angle \(\theta\) through which it has fallen, assuming the lower end does not move. Use these to show that the vertical component of force, where the rod touches the table, is \[N = mg(3\cos^2\theta - 2\cos\theta),\] and find the horizontal component. Let the coefficient of friction between the rod and the table be \(\mu\). Show that the rod's lower end either leaves the surface of the table when \(\cos\theta = \frac{1}{3}\), or slips when \(\tan\theta = \mu\). What determines which happens?
A smooth ring of elastic material (modulus of elasticity \(\lambda\)) has natural radius \(R\), negligible cross section, and mass \(M\). A smooth-sided right circular cone, whose vertex angle is \(2\alpha\), is held fixed with its axis vertical and vertex uppermost. The ring is placed over the cone so that it is always in contact with the cone, and moves so that the plane of the ring is always horizontal. If \(x\) is the distance between the centre of the ring and the vertex of the cone, show that the potential energy of the ring is, to within an additive constant, \begin{align*} \frac{\pi\lambda}{R}(x\tan\alpha - R)^2 - Mgx, \end{align*} where \(g\) is the acceleration due to gravity. By considering the total energy of the ring (or otherwise), find the equilibrium position. Show that when disturbed the ring oscillates about this position with simple harmonic motion, and find the period. [Modulus of elasticity \(\lambda\) is defined so that tension=\(\lambda \times\) (extension)\(\div\)(natural length).]
A particle of mass \(m\) is projected with velocity \(U\) horizontally and \(V\) vertically; gravity is constant with magnitude \(g\). Obtain the components of velocity as functions of time, and find the time of flight and the range to the point where it returns to its starting level. Slight air resistance, providing a force \(mk\) times the velocity, has now to be allowed for; \(k V/g\) is much less than unity. Approximating the vertical resistive force by using the velocity component found in (i), or otherwise, show that to first order in \(k V/g\) the time of flight is decreased by a fraction \(kV/3g\).
The motion of particles in the solar system, under the influence of the sun's gravity, is described by the equations (in appropriate units) \begin{align*} r - r\dot{\theta}^2 &= -1/r^2\\ r^2\dot{\theta} &= h = \text{const.} \end{align*} Using the second of these equations to give \(\theta\) as a function of \(r\), or otherwise, show that the first equation has the solution \begin{align*} \frac{1}{r} = \frac{1 + e\cos\theta}{h^2} \end{align*} for any constant \(e\). In the case \(0 \leq e < 1\), find the speed when the particle is nearest to the sun, and when it is furthest from it. A spaceship is in a circular orbit around the sun. Its velocity is increased instantaneously, parallel to itself, by a factor \(5/4\). Show that it will reach out to a distance \(25/7\) times its initial distance.
The real-valued functions \(x(t)\) and \(y(t)\) satisfy the pair of coupled differential equations \begin{align*} \ddot{x} + M\dot{y} - \omega^2 x &= 0\\ \ddot{y} - 2\omega\dot{x} - \omega^2 y &= 0 \end{align*} where \(\omega\) is a real constant, and dot denotes differentiation with respect to \(t\). Obtain a differential equation satisfied by \(x + \lambda y\) for a suitable choice of \(\lambda\), not necessarily real, and hence find the general solution of \((*)\). Describe briefly the shape of the path of the point \((x,y)\) when \(t\) is very large and positive.