\(ABCDE \dots\) is a closed polygon constructed of light rods \(AB, BC, \dots\) freely jointed at the vertices. It is in equilibrium under forces \(P_A\) applied at \(A\), and so on. A second (closed) polygon \(UVWXY \dots\) is constructed, in a similar manner, so that \(UV\) represents \(P_A\) (in magnitude and direction), \(VW\) represents \(P_B\), and so on. At \(V\) is applied a force represented in magnitude and direction by \(AB\), at \(W\) one represented by \(BC\), and so on all round the polygon. Prove that the second polygon is in equilibrium if and only if the lines of action of the forces \(P_A, P_B, \dots\) acting on the first polygon are concurrent.
Two fixed equally rough planes, intersecting in a horizontal line, are inclined at equal angles \(\theta\) to the vertical. A uniform rod rests between the planes, in a vertical plane at right angles to their line of intersection, and makes an angle \(\alpha\) with the vertical. If the rod is about to slip, show that \(\tan(\theta+\lambda) - \tan(\theta-\lambda) = 2 \cot \alpha\), where \(\lambda\) is the angle of friction. Deduce, or show otherwise, that if \(\alpha = \alpha_0\) in the position of limiting equilibrium then all positions with \(\alpha > \alpha_0\) are positions of equilibrium: and hence that if \(\mu\), the coefficient of friction, exceeds the positive root of the equation \(\mu^2 \sin^2 \theta + \mu \tan \theta - \cos^2 \theta = 0\), then all physically possible positions are positions of equilibrium.
A quadrilateral \(ABCD\) is formed from four uniform rods freely jointed at their ends. The rods \(AB\) and \(AD\) are equal in length and weight, and so also are the rods \(BC\) and \(CD\). The quadrilateral is suspended from \(A\) and a string joins \(A\) and \(C\) so that \(ABC\) is a right angle and \(BAD=2\theta\). Show that the tension in the string is \(w' + (w+w')\sin^2\theta\), where \(w\) is the weight of \(AB\) and \(w'\) is the weight of \(BC\).
If \(M(x)\) is the bending moment at a point distant \(x\) from one end of a thin straight horizontal beam of variable weight \(w(x)\) per unit length, show that, with suitable sign conventions, \(d^2M/dx^2 = w(x)\). A beam is required to satisfy the relation \(M(x)=k[w(x)]^2\), where \(k\) is a given constant. If one end of the beam is clamped and the other is free, obtain a differential equation satisfied by \(M(x)\), and hence show that the weight per unit length at distance \(x\) from the free end must be \(x^2/12k\). What are the dimensions of the constant \(k\), in terms of length, mass and time?
A train is running down a slope inclined at an angle \(\alpha\) to the horizontal, the engine exerting no tractive force. It is loading water at a rate \(p\) from a stream which is flowing at a speed \(u\) in the same direction as the train. Show that, if frictional resistance is neglected, the speed \(v\) of the train at time \(t\) is given by \[ (M_0+pt)v = (M_0t + \tfrac{1}{2}pt^2)g\sin\alpha + put + M_0v_0, \] where \(v_0, M_0\) denote the velocity and complete mass of the train at time \(t=0\).
Two masses, \(M\) and \(m\) (\(M>m\)), are connected by a string passing over a fixed smooth pulley. A cap of mass \(m'\), where \(m+m' > M\), is arranged so that whenever the mass \(m\) is at or above a certain height it must carry the mass \(m'\) with it (see Fig.~1), while when \(m\) falls below that height \(m'\) remains at rest on a stand. The mass \(m\) is pulled down a distance \(h\) from its equilibrium position and then released. If all impacts are inelastic find to what distance below its equilibrium position it will next return. Will the subsequent motion consist of (a) a finite number of oscillations followed by equilibrium, or (b) an infinite number of oscillations during some finite time, or (c) indefinitely prolonged oscillations?
A smooth hollow tube, in the form of an arc of a circle subtending an angle \(2(\pi-\theta)\) at its centre, where \(0<\theta<\frac{1}{2}\pi\), is fixed in a vertical plane with the ends of the tube uppermost and at the same horizontal level. Show that it is possible for a particle which fits the interior of the tube to perform continuous revolutions, leaving the tube at one end and re-entering it at the other. If the particle is of mass \(m\) and the arc of radius \(a\), find the velocity of the particle at the lowest point of its path, and the reaction between the particle and the tube at this point.
A see-saw consists of a smooth light frame \(ABC\) in the form of an isosceles triangle (\(AC=BC\)), freely pivoted at the mid-point \(D\) of \(AB\), and with a mass \(M\) attached at \(C\), where \(CD=h\). A particle of mass \(m\) is placed near \(D\). Assuming that when \(AB\) is inclined at \(\alpha\) to the horizontal the acceleration of the particle along \(AB\) is \(g\sin\alpha\), and that the amplitude of motion is so small that the reaction of the particle on the see-saw may be taken as \(mg\) perpendicular to \(AB\), show that (approximate) simple harmonic motion is possible, in which the oscillations of the particle along \(AB\) are in phase with those of the frame of the see-saw. Obtain equations giving the ratio of the amplitudes of these oscillations, and the frequency.
A uniform rod \(AB\) of length \(a\) and mass \(M\) is free to turn about a fixed point \(A\). A light rod \(BC\) also of length \(a\) is freely hinged at \(B\) to the rod \(AB\), and a particle of mass \(m\) is attached at \(C\). Initially \(ABC\) is a straight line, and the system is at rest on a smooth horizontal table. \(C\) is given an initial velocity \(v\) in a horizontal direction perpendicular to \(ABC\). When \(BC\) has turned through an angle \(\pi\) relative to \(AB\), \(C\) is suddenly brought to rest at \(A\). Determine the final angular velocity of the rod \(AB\), and the velocity of \(C\) immediately before it is brought to rest.
A uniform rod of mass \(m\) lying on a horizontal table is hit at its midpoint by a particle, also of mass \(m\), sliding along the table with velocity \(v\) perpendicular to the rod. Immediately after this impact one end of the rod hits an inelastic stop. If \(e\) is the coefficient of restitution between the particle and the rod, find the condition that the particle should immediately hit the rod again. If \(e\) is very small, will the final angular velocity of the rod be (i) practically the same as if the particle had coalesced with the rod, (ii) practically the same as if the rod had initially been free to turn about one fixed end?