A bead of mass \(m\) slides on a smooth wire in the form of an ellipse in a horizontal plane. It is attached to a particle of equal mass \(m\) by an inelastic string, which passes through a small smooth ring fixed at the centre of the ellipse. Initially the system is at rest in unstable equilibrium, with the bead at one end of the major axis of the ellipse: it is then slightly disturbed. For the instant when the bead passes through one end of the minor axis of the ellipse, find (i) the velocity of the bead, (ii) the tension in the string, (iii) the reaction of the wire on the bead.
A rigid wire is in the form of a semicircle of radius \(a\) with end points \(A\) and \(B\). Each element of the wire experiences a force tangential to the wire, the magnitude of the force on the element of length \(ds\) at \(P\) being equal to \(k\theta ds\) where, if \(O\) is the mid-point of \(AB\), \(\theta\) is the angle \(POA\) and \(k\) is a constant; the direction of the force is in the direction \(\theta\) increasing. The wire is held in equilibrium by two parallel forces acting respectively through \(A\) and \(B\). Determine the magnitudes and directions of these forces.
A beam of material of uniform density \(\rho\) is of the form of the solid of revolution obtained by the rotating of the straight lines \begin{align*} y &= a(-2 \le x \le -1), \\ y &= a(x+2) \quad (-1 \le x \le 0), \\ y &= a(2-x) \quad (0 \le x \le 1), \\ y &= a \quad (1 \le x \le 2) \end{align*} about the \(x\) axis. The beam rests on two supports at the same horizontal level at the sections \(x=-1, x=2\). Determine the bending moment and shearing force at the central section and give a rough sketch of the bending moment distribution along the beam.
A uniform circular cylinder of weight \(W\) and radius \(a\) rests on a rough horizontal plane. A uniform rod of length \(2l\) and weight \(w\) rests on the cylinder and the plane and lies in a plane normal to the generators of the cylinder. The coefficients of friction between the rod and the plane and the rod and the cylinder have the same value \(\mu_1\); the coefficient of friction between the cylinder and the plane is \(\mu_2\). Given that the rod starts to slip before the cylinder at an inclination \(\alpha\) to the horizontal where \(\cot \alpha/2 < 2l/a\), show that \[ \mu_1 = \tan \alpha/2, \quad \frac{l}{a} = \frac{1}{2}\sec\alpha \cot\alpha/2 \quad \text{and} \quad \mu_2 > \frac{w}{2W+w}\mu_1. \]
A rhombus \(ABCD\) is formed of four uniform \(AB, BC, CD, DA\) rods each of length \(a\) and weight \(w\) freely jointed together. The vertex \(A\) is fixed; the opposite vertex \(C\) is constrained to move along a vertical rail passing through \(A\), \(C\) being below \(A\). A light inextensible string is attached to \(C\), passes over a small pulley at \(A\) and carries a weight \(W>2w\). The vertices \(B\) and \(D\) are joined by a light elastic string of unstretched length \(\sqrt{2}a\) and modulus \(\lambda\). By considering the potential energy of the system, or otherwise, show that if \(\frac{(W-2w)}{\lambda}\) is small, the increase in length of the elastic string is approximately \[ \sqrt{2}a \left[ \frac{W-2w}{\lambda} - \frac{2w}{\lambda} + \frac{2(W-2w)^2}{\lambda^2} \right]. \]
A lift of mass \(M\) ascending vertically on frictionless guides is propelled by a motor of constant power \(R\). Starting from rest, power is maintained for a time \(t\) seconds and then shut off; the lift then comes to rest at a height \(h\) above its original position. Show that \(h=Rt/Mg\) and that the relation between total time of transit \(T\) between the two stops and the maximum velocity \(V\) is \[ T = \frac{R}{Mg^2} \log_e \frac{R}{R-MgV}. \]
A hostile aircraft is flying a horizontal course with uniform speed \(U\) ft./sec. at height \(h\) feet. The course passes vertically above a gun site; the crew receives warning of its approach but is unready to fire until the instant when the aircraft is vertically overhead. The muzzle velocity of shells fired from the gun is \(V\) ft./sec. and it may be assumed that \(V^2 > 2U^2+2gh\). Neglecting air resistance, determine the time interval throughout which the aircraft is in danger from this gun assuming that shots on the descending branch of a trajectory are dangerous as well as those on an ascending branch.
Show that the potential energy of a light string of unstretched length \(a\) and modulus \(\lambda\) is \(\frac{\lambda}{2a}(x-a)^2\) when its length is \(x\) (\(>a\)). A bead of mass \(m\) can move freely along a smooth parabolic wire of latus rectum \(4a\) the plane of the wire being horizontal. A light elastic string of modulus \(\lambda\) and unstretched length \(a\) is attached to the bead the other end being fixed at the focus of the parabola. Show by energy considerations, or otherwise, that if the bead is released from rest at the end of the latus rectum it reaches the vertex in a time \(\pi\sqrt{\frac{ma}{\lambda}}\) and has then a velocity \(\sqrt{\frac{\lambda a}{m}}\).
A particle can move freely on a horizontal table inside a circular barrier of radius \(a\) formed by a circular cylinder fixed to the table with its axis vertical. The particle is projected with velocity \(V\) along the table from a point \(A\) of the barrier along a chord \(AB\) subtending an angle \(2\alpha\) (\(<\pi\)) at the centre. The coefficient of restitution between the particle and the barrier is \(e\). Show that the particle ultimately reaches a steady state of motion circulating uniformly round the inside of the barrier in a time less than \(\frac{2a\sin\alpha}{V \cos^2\alpha}\frac{1}{1-e}\). What is the velocity of this circular motion? Discuss briefly the case \(\alpha = \pi/2\).
A bead can move freely on a smooth rigid wire in the form of an ellipse of semiaxes \(a\) and \(b\) (\(a>b\)). The wire is made to rotate about the minor axis of the ellipse which is vertical with constant angular velocity \(\omega\). If \(gb < a^2\omega^2\), show that there are four positions of relative equilibrium of which two are given by eccentric angles \(\pi + \sin^{-1}\frac{gb}{a^2\omega^2}\), \(2\pi - \sin^{-1}\frac{gb}{a^2\omega^2}\). Show further that these two positions are stable.