A uniform solid circular cylinder of radius \(a\) with its axis horizontal is lying on a rough horizontal plane, the coefficient of friction being \(\mu\). The cylinder is projected horizont- ally with velocity \(V\) at right angles to its axis, and at the same time is given an angular velocity \(\Omega\) about its axis. The cylinder is required to come to rest at a distance \(d\) from its starting point. Find whether this is possible, and if so what the values of \(V\) and \(\Omega\) must be.
Define the centre of mean position of \(n\) points \(P_1, P_2, \dots, P_n\) in a plane (centre of gravity of equal masses at \(P_1, P_2, \dots, P_n\)). Prove that the point so defined is independent of the axes of reference used. A system of forces in a plane acts on a rigid body. \(P_1, P_2, \dots, P_n\) are \(n\) points in the plane, and their centre of mean position is \(P_0\). If the moment of the system of forces about \(P_r\) is denoted by \(M_r\), prove that \[ M_1+M_2+\dots+M_n = nM_0. \]
A uniform rod \(ABCDE\), of length \(6a\) and weight \(W\), rests on two supports at the same level at \(B\) and \(D\), where \(AB\) and \(DE\) are each of length \(a\); a weight \(W/4\) is hung from the middle point \(C\) of the rod, and weights each \(W/8\) are hung from the ends \(A\) and \(E\). Find the bending moment \(M\) at a point on \(AC\) at distance \(x\) from \(A\), and illustrate by a graph. Shew in particular that \(M\) vanishes when \(x=3a/2\), and determine the greatest value of \(|M|\).
Two uniform rods \(AB, BC\), each of length \(2\sqrt{2}a\) and weight \(W\), are smoothly hinged together at \(B\). The end \(A\) is freely attached to a fixed point of a horizontal rail, and the end \(C\) is freely attached to a small light ring which slides on the rail. The coefficient of friction between the ring and the rail is \(2/3\). Prove that the system can rest in equilibrium (with \(B\) below the rail) with the rods at right angles. When the system is in this position the rail is slowly tilted, with \(C\) at a lower level than \(A\). Find the inclination of the rail to the horizontal when equilibrium is broken.
A bead of mass \(m\) slides on a smooth circular hoop of radius \(a\) which is fixed in a vertical plane. The bead is attached to one end of a light inextensible string which passes through a small smooth ring fixed at a height \(b\) (\(>a\)) vertically above the centre of the hoop; the other end of the string is attached to a particle of mass \(m\) hanging freely. Prove that when the bead is at an angular distance \(\theta\) from the highest point of the hoop the potential energy of the system is \[ mg(a\cos\theta+r) + \text{constant}, \] where \[ r^2 = a^2+b^2-2ab\cos\theta. \] Find the positions of equilibrium of the system, and discuss their stability.
An inelastic string \(AC\), whose mid-point is \(B\), has variable line-density, the line-density at two points equidistant from \(B\) being the same. The ends \(A, C\) are held at the same level, and the string hangs at rest under gravity. The tangent at a point \(P\) of the string makes an angle \(\psi\) with the horizontal; the arc \(BP\) has length \(s\) and weight \(W\). Prove that \[ W=T_0\tan\psi, \] where \(T_0\) is constant. If the line-density is proportional to \[ \frac{a^2}{a^2+s^2}, \] where \(2a\) is the length of the string, prove that the form in which the string hangs satisfies the equation \[ s=a\tan\left(\frac{\pi\tan\psi}{4\tan\alpha}\right), \] where \(\alpha\) is the value of \(\psi\) at \(C\).
An engine is required to raise a weight of 1 ton from the bottom of a mine 900 feet deep in 5 minutes. What must be the average horse-power of the engine? What must it be if the load, starting from rest at the bottom, has a velocity of 12 feet per second on reaching the surface?
A particle of mass \(m\) moves on a straight line under an attraction towards a fixed point \(O\) of the line, of magnitude \(mn^2\) times the distance from \(O\). Determine the position of the particle at time \(t\) if it is projected from \(O\) with velocity \(u\) at the instant \(t=0\). Draw a graph shewing the relation between \(t\) and \(x\) for different values of \(u\). Shew that \(u\) can be chosen so that the particle passes through the given point \(x=a\) when \(t=t_1\), provided that \(t_1\) is not an integral multiple of \(\pi/n\): why are such values of \(t_1\) exceptional?
A simple pendulum consists of a particle of mass \(m\) attached to a fixed point \(O\) by a light inelastic string of length \(a\). The particle moves in a complete vertical circle in such a way that the tension in the string just vanishes at the highest point. What is the tension at the lowest point? Prove that the greatest value of the horizontal component of the tension during the motion is \(9\sqrt{3}mg/4\).
At the instant \(t=0\) particles are projected horizontally, in a given vertical plane, from different points of a vertical straight line, the velocity of projection being that due to fall from a fixed point \(O\) of the line. Prove that the paths of the particles are a family of parabolas, and that at any instant all the particles lie on one member of the family. What is the envelope of the family?