Problems

A heavy uniform inelastic string of length \(l\) has one end attached to a fixed peg, and passes through a smooth ring distant \(2h\) from the peg and on the same horizontal level. Shew that the length \(y\) of string which hangs vertically from the ring is given by \[ \frac{l-y}{2y} = \tanh \frac{2h}{\sqrt{\{(3y-l)(y+l)\}}}. \]

A bead of mass \(m\) is free to move on a smooth rod which is constrained to rotate about one end in a vertical plane with constant angular velocity \(\omega\). Initially the bead is at rest at the fixed end of the rod, and the rod is pointing vertically downwards. Find the differential equation satisfied by the distance \(r\) moved by the bead along the rod. Deduce the value of \(r\) at time \(t\), and shew that the reaction between the bead and the rod at that instant is \(mg(\sinh\omega t + 2\sin\omega t)\).

A light string of natural length \(2l\) and modulus of elasticity \(mg\) is attached to two points at a distance \(2\sqrt{3}l\) apart on the same horizontal level. A small stone of mass \(m\) is held against its middle point as for a catapult, and then depressed a distance \(3l\) vertically downwards. Shew that the stone rises when released. Shew also that it is projected (the stone and the string being in contact till the horizontal position is reached) to a distance \(2(3-\sqrt{3})l\) above the two fixed points.

A shell is projected vertically upwards from the ground, its kinetic energy initially being \(E\). When its velocity has been reduced by one half, it explodes into two parts of masses \(M_1\) and \(M_2\), of which \(M_1\) is the upper. The kinetic energy just after the explosion exceeds that just before by \(E\), and the explosion acts along the line in which the shell is travelling. Shew that the upper portion reaches a height \[ \frac{M_1+M_2+\sqrt{M_1 M_2}}{M_1(M_1+M_2)} \frac{E}{g} \] above the ground.

A particle of mass \(m\) is at the centre of the base of a smooth rectangular box of mass \(M\) which rests on a smooth horizontal table. The particle is projected along the base so as to meet normally the face P of the box. The coefficient of restitution of the particle and either the face P or the opposite face is \(e\). The velocity of the particle and of the box respectively, measured in the direction and sense of the initial velocity, just before the \(n\)th impact with the face P are denoted by \(u_n\) and \(V_n\) respectively; where \(v_1\) is the initial velocity \(v\) of the particle, and \(V_1\) is zero. By considering the values of the functions \(\lambda_n = u_n - V_n\) and \(\mu_n = MV_n + mu_n\), evaluate \(V_n\) and \(u_n\), and shew that they tend to a common limit, which is the value which they would have had if the particle had stuck to the box on its first impact.

A lamina of uniform density \(\rho\) is free to turn about an axis in its own plane through the centre of mass. It is acted on by a constant couple whose moment about the axis is \(L\). The resistance of the atmosphere produces at every point of the lamina a resisting normal pressure which, per unit area, is \(k\) times the velocity. Shew that the equation of motion is \[ I\ddot{\theta} = L - (kI/\rho)\dot{\theta}, \] where \(I\) is the moment of inertia about the axis of rotation. If the lamina is initially at rest, find its angular velocity at time \(t\).

A uniform circular disc of radius \(a\) and mass \(m\) rolls without slipping in a vertical plane on a horizontal table. Its centre is attached by two elastic strings each of natural length \(l\) and modulus of elasticity \(\lambda\), to two fixed points, one on either side of it, each at a height \(a\) above the table and in the same vertical plane as the disc. Assuming that each string remains taut throughout the motion, shew that the disc, if slightly disturbed from its position of equilibrium, executes a simple harmonic motion of period \(\pi\sqrt(3ml/\lambda)\).

Two circular cylinders, of radii \(2a, 3a\) respectively, are fixed rigidly to a horizontal plane. One generator of each cylinder is in contact with the plane and each cylinder touches the other along a generator. A uniform rod of length \(l\) rests in equilibrium in contact with both cylinders and perpendicular to the axes of the cylinders, and its lower end rests against a vertical wall which is parallel to the axes of the cylinders. If all the surfaces are smooth and if the reactions between the rod and the cylinders are equal, obtain an expression for the distance of the wall from the axis of the smaller cylinder and deduce that \(l\) must lie between \(8.9a\) and \(28.9a\) approximately.

A heavy elastic string of natural length \(2\pi a \cos\beta\) rests in equilibrium round a horizontal small circle (of radius \(a\cos\alpha\)) of a rough solid sphere of radius \(a\). The string is higher than the centre of the sphere and is on the point of sliding upwards. \(\beta > \alpha\). Prove that the coefficient of elasticity of the string is \[ \frac{mga \cos\alpha \cos\beta \cot(\alpha-\lambda)}{\cos\alpha - \cos\beta}, \] where \(m\) is the mass of the string per unit length in the stretched state, and \(\lambda\) the angle of friction between the string and the sphere.

Forces of magnitudes \(\lambda_1.OP_1, \lambda_2.OP_2, \dots \lambda_n.OP_n\) act on a particle at \(O\) along the lines \(OP_1, OP_2, \dots OP_n\) respectively. Prove that their resultant acts along \(OG\) and is of magnitude \((\lambda_1 + \lambda_2 + \dots + \lambda_n)OG\), where \(G\) is the centre of mass of masses proportional to \(\lambda_1, \lambda_2, \dots \lambda_n\) placed respectively at \(P_1, P_2, \dots P_n\). A particle \(O\) is maintained in equilibrium on a smooth horizontal table by three elastic strings. The strings have coefficients of elasticity \(\lambda, 2\lambda, 3\lambda\) respectively and pass one through each of three smooth rings fixed to the table at the vertices \(A, B, C\) of an equilateral triangle of side \(l\sqrt{3}\). The other end of each string is fixed at the centre of the triangle \(ABC\) and each string has unstretched length \(l\). Find the distances of \(O\) from \(A, B, C\) in terms of \(l\).