A sphere moving with velocity \(\mathbf{u}_1 = a_1\mathbf{u}\) collides with a similar sphere moving with velocity \(\mathbf{u}_2 = a_2\mathbf{u}\). Momentum and energy are conserved in the collision, after which the spheres have velocities \(\mathbf{v}_1\) and \(\mathbf{v}_2\) respectively. Show that if \(\mathbf{v}_1\) and \(\mathbf{v}_2\) are mutually perpendicular then one of the spheres must initially have been stationary. Is the converse true? If both spheres have the same speed \(c|\mathbf{u}|\) after the collision, show that \(c^2 \cos \theta = a_1 a_2\), where \(\theta\) is the angle between \(\mathbf{v}_1\) and \(\mathbf{v}_2\).
A hollow circular cylinder of moment of inertia \(I\) about its axis is initially at rest. It is made to spin about its axis by a motor which applies a constant torque \(G\). The motion is opposed by a frictional torque \((G/\omega_0)\omega\), where \(\omega\) is the angular velocity of the cylinder. Find \(\omega\) as a function of time and show that it tends to a limiting value. When the cylinder is rotating at this limiting rate a particle (whose mass is so small that its effect on the motion of the cylinder is negligible) moves on the inner surface of the cylinder, in a plane perpendicular to the axis, with an initial angular velocity \(2\omega_0\) about the axis. The coefficient of friction is \(\mu\). How much time elapses before the particle rotates at the same rate as the cylinder? [The force of gravity may be neglected.]
An earth satellite experiences a gravitational acceleration \(-\gamma r/r^3\), where \(\mathbf{r}\) is its position vector relative to the centre of the earth. Find the period of a satellite moving in a circular orbit of radius \(r_0\). A space vehicle ejected from this satellite acquires initially an additional radial velocity \(v_0\). Obtain the maximum distance from the earth reached by this vehicle. What is the least value of \(v_0\) for which the space vehicle escapes to outer space?
In the electric circuit below, the charge \(Q\) on the capacitor \(C\) is related to the applied electromotive force \(E\) by the differential equation $$L \frac{d^2Q}{dt^2} + 2R \frac{dQ}{dt} + Q/C = E,$$ where \(L > R^2C\), and \(E(t) = E_0 \cos \omega t\), where \(E_0\) and \(\omega\) are constants. Show that the current \(I(t) (= dQ/dt)\) is ultimately in phase with \(E(t)\) if and only if \(\omega^2 LC = 1\).