A corner is sawn off a uniform cube. The plane of the cut is equally inclined to the three edges it meets, and is distant \(h\) from the vertex of the cube. Find the distance of the centre of mass of the smaller piece from the vertex. Another corner is sawn off; this time the plane meets the edges at distances \(a\), \(b\), \(c\) from the nearest vertex. What can you say about the position of the mass-centre of this piece?
\(A\) and \(B\) are two small islands in an estuary; \(B\) is at a distance \(a\) to the north of \(A\). A motor-boat, which would travel at a speed \(v\) in still water, goes directly from \(A\) to \(B\) and back. Obtain a formula showing how the total time taken depends on the speed \(u\) of a uniform current in the water and on its direction. Given that \(a = 850\) ft, \(v = 9\) ft/s, the total time is 700 s, and the velocity of the water is 8 ft/s, what can you say about its direction?
A satellite in the form of a large right circular cylinder, of radius \(a\), is moving with velocity \(v\) in a direction parallel to its axis in a medium containing small particles each of mass \(m\) and coefficient of restitution \(e\). The particles are initially at rest. Show that when the number \(n\) of particles per unit volume is very large the satellite is in effect subject to a retarding force \(\pi(1+e)nma^2v^2\). Without making any calculations, discuss whether the apparent retarding force would be greater or less, if the satellite were a sphere of radius \(a\).
Four particles \(A\), \(B\), \(C\), \(D\), each of mass 1, are connected by light rods \(AB\), \(BC\), \(CD\), \(DA\) to form a square. \(A\) is attached to a fixed point, and the system hangs from it, with a thread \(AC\) maintaining the square in shape. The thread is then cut. Find the acceleration with which \(C\) begins to descend.
A train of mass \(m\) is driven by electric motors which exert a force. The force depends linearly on the velocity, decreasing to zero at a speed \(V\), and its resistance to its motion is \(m(2v/3V)^2\) at any speed \(v\). Find how far it has travelled from rest when it attains a given speed \(V\).
A smooth wedge of mass \(M\) stands on a smooth horizontal table. A particle of mass \(m\) is placed on the wedge, at a height \(h\) above the table, and slides down. A particle reaches the greatest slope, which is inclined at an angle \(\beta\) to the horizontal and passes through the vertical plane as the mass centre of the wedge. Find how far the wedge has moved when the particle reaches the bottom of the slope. Find also the time taken.
Two small rings \(P\) and \(Q\) can slide on a fixed horizontal wire \(OPQ\). The ring \(P\), of mass \(m\), is connected with \(P\) by a light spring of natural length \(l\), which exerts a force \(m\omega^2(l-i)\) when its length is \(i\). \(P\) is now attached by a rod of length \(a\) to a fixed vertical wire through \(O\). \(R\) is made to oscillate about \(O\) so that its displacement is \(s = h\sin\omega t\). Find a formula describing the possible motions of \(Q\), assuming that \((h/a)^2\) can be neglected. Comment, without calculations, on any exceptional case.
A smooth tube of length \(2a\) is constrained to rotate in a horizontal plane about its centre \(O\) with constant angular velocity \(\omega\). A particle in the tube is projected with velocity \(u\) from \(O\). Find its speed immediately after it leaves the tube. Find also the force acting on it while still in the tube, at a time \(t\) after it has left \(O\). Verify that the kinetic energy gained by the particle is equal to the work done in keeping the angular velocity of the tube constant.
A fixed hollow sphere of radius \(a\) has a small hole bored through its highest point, resting on the inside of the sphere; there is no friction. Find, for any value of \(b/a\), how many positions of equilibrium there are with the rod in a given vertical plane and which of them are stable.
Find the moment of inertia of a uniform cube of side \(2a\) about one edge. The cube is released from rest on a smooth plane at an angle \(\theta\) to the horizontal. After sliding down a distance \(b\), it meets a small inelastic ridge \(A\). Find the angular velocity with which it begins to turn about \(A\). Show that it will not get over the obstacle unless $$3b\sin\theta \geq 16\sqrt{2} \cdot a\sin^2\left(\frac{1}{4}\pi - \frac{1}{2}\theta\right).$$