Problems

A heavy uniform chain of line density \(w\) hangs over a rough circular cylinder of radius \(a\) having its axis horizontal and perpendicular to the plane of the chain. Obtain in the case of limiting equilibrium the equation involving the value of the tension \(T\) at a point of the chain in contact with the cylinder, in the form \[ \frac{dT}{d\theta} - \mu T = wa\sec\lambda\cos(\theta-\lambda), \] where \(\mu=\tan\lambda\) is the coefficient of friction and \(\theta\) is measured from the point at which the shorter end of the chain comes into contact with the cylinder. Hence prove that if the shorter free length is just zero, the greatest length possible on the other side is \(a\sin 2\lambda(1+e^{\mu\pi})\).

The ends of a heavy uniform rod of length \(a\) are constrained by rings to move on a rough circular wire of radius \(a\) fixed in a vertical plane. If \(\mu\), the coefficient of friction, is less than \(\sqrt{3}\), shew that the greatest inclination, \(\theta\), of the rod to the horizontal when in equilibrium is \(\tan^{-1}\dfrac{4\mu}{3-\mu^2}\).

A regular pentagon \(ABCDE\) consists of heavy uniform rods each of weight \(W\) freely jointed at their ends. The whole rests in a vertical plane with \(AB\) in contact with a smooth horizontal table, rigidity being maintained by two light struts \(AD, BD\). Calculate the thrust in these struts.

A bead is free to move on a smooth straight wire rotating in a horizontal plane about a given point of itself with constant angular velocity \(\omega\). Find the equation of motion of the bead if released from rest at a point of the wire, and shew that the path is a spiral whose polar equation can be expressed in the form \(r=a\cosh\theta\). Shew also that the velocity \(v\) of the particle in any position is given by \(v^2 = a^2\omega^2\cosh 2\theta\), and that \(r\) doubles its initial value in a time which is \(\cdot 2096\) of the time of a complete revolution.

A particle is projected under gravity and moves in a medium which offers resistance to motion equal to \(KV\) per unit mass, where \(V\) is the velocity. Shew that the slope of the velocity at any subsequent instant differs from a certain fixed slope by an amount \(\dfrac{g}{uK}e^{Kt}\), where \(u\) is the horizontal component of initial velocity. Shew also that the fixed slope in question is the direction of acceleration.

A particle is to be projected with given velocity in a vertical plane from a certain horizontal level so that it will surmount an obstacle of height \(h\) and pass under another obstacle at a height \(k\) at a distance \(d\) beyond the former. Describe and justify a graphical construction for obtaining the maximum distance beyond the obstacles it is possible to reach.

A uniform circular cylinder of radius \(a\) rests on a rough horizontal plane. A horizontal blow is delivered in a vertical plane through its centre of gravity and at a height \(\frac{3}{2}a\) above the ground. Neglecting impulsive frictional force, shew that slipping ceases when the linear velocity of the cylinder is \(\frac{5}{9}\) of its original instantaneous value.

A thin heavy flexible chain of mass \(M\) and length \(l\) is wound round a cylindrical drum of radius \(a\) and moment of inertia \(I\) about its axis which is vertical and round which it can turn freely. The inner end of the chain is attached to the drum, while the free end rests on a small smooth pulley of negligible distance from the surface of the drum and will fall vertically when the drum revolves. The drum is given an initial peripheral velocity \(u\) and the chain unwinds without slipping. Shew that the chain will have unwound completely in a time \[ t=\frac{1}{n}\log_e\left[\frac{nl}{u} + \sqrt{1+\frac{n^2l^2}{u^2}}\right], \quad \text{where } n^2 = \frac{Mg}{I(\frac{1}{a^2}+M)}. \]

A particle of mass \(3m\) is suspended by a light inextensible string of length \(l\) from a body of mass \(m\) which can move freely on a horizontal rail. If the lower particle is released from rest with the string taut and inclined at a small angle \(\alpha\) to the vertical, determine the amplitudes and the periods of the resulting small oscillations.

Shew that \[ 5\{(y-z)^7 + (z-x)^7 + (x-y)^7\} = 7\{(y-z)^5+(z-x)^5+(x-y)^5\}\{x^2+y^2+z^2-yz-zx-xy\}. \] Hence, or otherwise, find all the factors of \[ (y-z)^7 + (z-x)^7 + (x-y)^7. \]