Weights \(w_i\) (\(i = 1, 2, \ldots, n\)) are hung from points of a light inextensible string which is suspended at its two ends from given fixed points. The lengths of the segments are given. Show how to obtain sufficient equations to determine the tension and inclination of each segment. Equal weights \(W\) are attached at equal horizontal intervals \(a\) to a light inextensible string which hangs between given points. Show that the points of attachment of the weights lie on a parabola of latus rectum \(2aW/H\), where \(H\) is the horizontal component of tension.
A uniform heavy rod \(AB\) of length \(2l\) can turn freely about a fixed point \(A\), and \(C\) is a fixed point at height \(2a\) vertically above \(A\). A small heavy ring of weight equal to that of the rod can slide smoothly along the rod, and is attached to \(C\) by a light inelastic string of length \(a\). Given that \(a < 2l\), prove that the configuration in which the rod is vertical is one of stable equilibrium if \(a > l\).
A uniform heavy string hangs in a vertical plane over a rough peg which is a horizontal cylinder of circular cross-section whose axis is perpendicular to the plane. The radius of the cylinder is \(a\) and the coefficient of friction is \(\mu\). If one free end of the string lies at a point of the cross-section where the tangent is vertical, prove that the greatest length of string which can hang vertically on the other side of the peg is \(2\mu a(1 + e^{\mu\pi})(\mu^2 + 1)\).
An aeroplane, which would fly with speed \(V\) in still air, flies in a wind of uniform velocity \(kV\), where \(0 \leq k < 1\). The aeroplane is steered so that it moves relative to the ground in a direction inclined at an angle \(\psi\) to the wind. Find the speed of the aeroplane relative to the ground in terms of \(V\), \(k\) and \(\psi\). Show that the time required to fly around a circuit fixed relative to the ground in a uniform wind is \[ \frac{1}{(1-k^2)} \oint \frac{\{(1-k^2\sin^2\psi) - k\cos\psi\} ds}{V} \] where \(ds\) is an element of length along the circuit and the integral is taken round the circuit. Hence prove that this time lies between \(T/\sqrt{(1-k^2)}\) and \(T/(1-k^2)\), where \(T\) is the time required in still air.
A small animal of mass \(m\) stands on the horizontal floor of a truck of mass \(M\) which is free to move on horizontal rails. The animal jumps (from rest) in a vertical plane parallel to the rails so as just to clear the vertical tailboard, which is at distance \(a\) from it and of height \(h\) from the floor. Show that the impulsive reaction between the animal and the truck has least magnitude when it makes with the horizontal an angle \(\alpha\), where \[ \tan 2\alpha = -Ma/(M + m)h. \]
Two particles \(P_1\) and \(P_2\) of masses \(m_1\) and \(m_2\) respectively are connected by a light inextensible string. \(P_1\) lies on a smooth horizontal table, the string passes through a small hole \(O\) in the table, and \(P_2\) hangs below the table. Initially \(P_1\) is at distance \(a\) from \(O\) and moves at right angles to the radius \(OP_1\) with speed \(V\). In the subsequent motion the distance \(OP_1\) at time \(t\) is \(r\). Obtain an equation for this motion in the form \(r^3 = f(r, a, V)\). Show that if at any subsequent instant \(P_1\) again moves at right angles to \(OP_1\), then at that instant \(r\) must equal \(a\) or \(l + \sqrt{(l^2 + 2al)}\), where \(l = m_1V^2/m_2g\).
Starting from the equation of motion of a single particle, develop the dynamical theory of the motion of a system of particles, including the equations of linear and angular momentum. Indicate how to derive the theory of motion of a rigid body from the dynamics of a system of particles.
A uniform rod \(AB\) of length \(2a\) and mass \(m\) stands balanced vertically on a smooth horizontal table, \(A\) being the point of contact. A horizontal impulse \(I\) is applied at \(A\). Show that if \(I > \frac{1}{2}m\sqrt{(ag)}\) the rod leaves the table immediately, and in this case find that value of \(I\) for which the rod is horizontal at its first impact with the table.
An inaccessible vertical tower \(CD\) of height \(h\) is observed from two points \(A\) and \(B\) which lie on a horizontal straight line \(ABC\) through the base \(C\). The distance \(AB\) is \(a\) and the elevation of \(D\) is \(\alpha\) from \(A\) and \(\beta\) from \(B\). Find an expression for the distance \(BC\) in terms of \(a\), \(\alpha\) and \(\beta\). If small errors \(\pm \epsilon\) may be made in observing each of \(\alpha\) and \(\beta\), show that the greatest proportional error in \(BC\) is \[ \frac{\epsilon\sin(\beta+\alpha)}{\sin\alpha\cos\beta\tan(\beta-\alpha)}. \]
State conditions for two plane distributions of matter to be equimomental. Prove that a uniform triangular plate of mass \(M\) is equimomental with the system of three masses each \(\frac{1}{3}M\) at the mid-points of the sides of the plate. A uniform triangular plate has edges of length \(a\), \(b\), \(c\). Prove that the radius of gyration of the plate about an axis through its centroid normal to its plane is \[ \frac{1}{6}\sqrt{(a^2+b^2+c^2)}. \]