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?
Forces \(P, Q, R\) and \(S\) act in the sense indicated along the sides \(AB, BC, CD, DA\) of a square \(ABCD\) whose sides are of length \(l\). Reduce this system to
A square table of weight \(W\) has side \(2a\) and height \(b\). The top is uniform and it has four equal legs at its corners. It stands on a rough horizontal floor of coefficient of friction \(\mu\). Given that \[ \frac{b}{a} < \frac{1-\mu^2}{\mu}, \] find the magnitude and elevation of the least force that will make the table slide on the floor parallel to a side. Show also that this force may be applied at any point of the table without toppling it.
A uniform heavy beam of length \(2l\) and weight \(2W\) rests on two supports at distance \(\frac{1}{4}l\) from either end. Weights of magnitude \(W\) are suspended from each end and an upward force \(2kW\) (\(k<2\)) is applied to the mid-point of the beam. Derive expressions for the shearing force and the bending moment at a general point of the beam and prove that the numerically greatest bending moment occurs at the supports if \(k<\frac{5}{8}\) and at the mid-point if \(k>\frac{5}{8}\).
A light inextensible string is wound a number of times round a horizontal circular cylinder. The two ends of the string hang vertically downwards supporting weights \(w\) and \(32w\) respectively. The coefficient of friction between the string and the cylinder is \(\dfrac{1}{\pi}\log_e 2\). Derive the equation for the variation of the tension along the string when the string is on the point of slipping, and find the least number of turns of the string round the cylinder for the system to be in equilibrium.
A sphere of radius \(a\) is intersected by a plane at a distance \(\frac{1}{2}a\) from its centre. A solid of revolution consists of the larger of the two parts into which the sphere is so divided together with the right circular cylinder of radius \(\frac{\sqrt{3}}{2}a\) and height \(\frac{1}{2}a\) whose base coincides with the plane section of the sphere. The density of the whole solid is uniform. Find the mass centre of the solid and the moment of inertia about the axis of symmetry.