When it is on level ground, the centre of gravity of a motor car is at height \(h\) and its front and rear axles are at horizontal distances \(a\) and \(b\) from the centre of gravity. If it is parked facing up a slope which makes an angle \(\alpha\) with the horizontal, with its rear wheels locked, show that the coefficient of friction between the rear wheels and the road must not be less than \((a + b)/(h + a \cos \alpha)\). What is the corresponding result if it is parked facing down the slope?
When a cyclist travels due E. with speed \(U_1\), the wind appears to come from a direction \(\alpha\) E. of N.; when his speed is \(U_2\) due N. its apparent direction is \(\beta\) E. of N. What is the strength of the wind and the direction from which it is actually blowing? If the cyclist now travels N.E. with speed \(U_3\), what is the strength of the head-on component of apparent wind velocity?
A particle is initially describing a circular orbit under an attractive force \(\mu/r^n\) (per unit mass), where \(r\) is the distance from the centre of force, \(\mu\) is a constant and \(n\) is a number greater than unity. The particle is then acted on by a retarding force for a finite time, and after it has ceased to act the particle is again describing a circular orbit about the centre of force. Is the radius greater or smaller than it was originally?
A boy wishes to kick a ball through a window which is at horizontal distance \(l\). The bottom of the window is at height \(h\) and the top at height \(h + a\). He kicks the ball with a velocity \(V\) at elevation \(\alpha\). Regarding the ball as a point particle and neglecting forces other than gravity, find (i) the maximum value \(V_0\) of \(V\) for which the ball cannot pass through the window for any value of \(\alpha\); (ii) for \(V > V_0\), the values of \(\alpha\) for which the ball passes through the window.
A circular groove of radius \(a\) is marked out on a plane inclined at an angle \(\alpha\) to the horizontal. A particle is projected along the groove, from its lowest point, with velocity \(V_0\). For all values of \(V_0\), find in terms of a definite integral the time that elapses before the particle is again at the lowest point of the groove. Show that, if \(V_0\) is large, the time is approximately $$\frac{2\pi a}{V_0}\left[1 + \frac{g a}{4 V_0^2} \sin^2 \alpha\right].$$ (Frictional forces are to be neglected and it may be assumed that the particle does not leave the groove.)
A rocket of initial total mass \(M_0\) (including fuel \( < M_0\)) moves vertically under gravity in a resisting medium. The resisting force, per unit mass, is a function \(f(V)\) of the velocity \(V\) which vanishes when \(V = 0\). The fuel is ejected from the rocket with a constant velocity \(U\) relative to the rocket and the rate of burning at any time is proportional to the total mass of rocket and fuel remaining \((dM/dt = -\lambda M)\). Show that the rocket cannot begin to descend until after the fuel is exhausted. If \(f(V) = kV\), where \(k\) is a constant, find the height reached at the moment of fuel exhaustion.
The elastic strings \(AB\), \(BC\) have unstretched lengths \(l\) and moduli of elasticity \(3\lambda mg\) and \(2\lambda mg\) respectively. \(A\) is attached to a fixed support and particles of mass \(m\) are attached at \(B\) and \(C\) and the system hangs in equilibrium vertically. The particles at \(B\) and \(C\) are now displaced vertically downwards through distances \(x_0\) and \(y_0\) respectively from their equilibrium positions and are then released. If the subsequent displacements of the particles from their equilibrium positions are \(x\) and \(y\), show that \(x + 2y\) and \(2x - y\) vary harmonically with time and find their periods. If \(y_0 = 2x_0\), find expressions for \(x\) and \(y\) as functions of the time. (It may be assumed that \(x_0\) and \(y_0\) are so small that neither string ever becomes slack.)
A locomotive working at constant power \(P\) draws a total load \(M\) against a constant resistance \(R\). It starts from rest. Show that the distance travelled in reaching the speed \(v_1\) is $$\frac{Mv_0^2}{R}\left[\log\frac{v_0}{v_0 - v_1} - \frac{1v_1^2}{v_0 2v_0^2}\right],$$ where \(v_0 = P/R\). What is the time taken to reach the speed \(v_1\)?
Define the moment of inertia of a solid body about an axis and state and prove the 'parallel axis' theorem. A solid body is formed by rotating the square \(ABCD\) of side \(a\) about its diagonal \(AC\). Find the moment of inertia of the body (assumed to be of uniform density) about the axis through \(D\) parallel to \(AC\).
Find the moment of inertia of a uniform thin rod of length \(2a\) and mass \(m\) about an axis perpendicular to it through one end. Such a rod rotates, with angular velocity \(\Omega\), on a smooth horizontal table about an axis through one end which is fixed. If the rod strikes a stationary particle of mass \(m\) at distance \(t\) from the fixed end and the particle sticks to the rod, what is the new value of the angular velocity? How much heat is generated by the impact?