A particle slides down the outside of a fixed smooth sphere of radius \(r\), starting from rest at a height \(\frac{1}{8}r\), measured vertically, above the centre. Prove that it leaves the sphere when at a height \(\frac{1}{4}r\) above the centre. Prove also that when the particle is at a horizontal distance \(r\sqrt{2}\) from the centre, it is at a vertical depth \(4r\) below the centre.
If the tractive force per ton of an electric train at speed \(v\) is \[ \frac{a(b-v)}{c+v} \text{ tons weight}, \] where \(a, b\) and \(c\) are constants, find the speed \(V\) at which the horse-power exerted is a maximum. Find also the gradient up which, if friction and wind resistance are neglected, the maximum speed attainable is \(V\). \item[(i)] If \[ x^m y^n = (x+y)^{m+n}, \] prove that \[ \frac{dy}{dx} = \frac{y}{x}. \] \item[(ii)] Sketch roughly the shape of the curve \[ y^2 = x(x-1)(2-x), \] and prove that part of it is an oval of breadth 1, and depth \(\sqrt{\frac{4}{27}}\). Note: The scanned document depth value is hard to read. It's likely `sqrt(4/27)` or `sqrt(64/27)`. Using the value that seems more correct.
If a tree trunk \(l\) feet long is a frustum of a cone, the radii of its ends being \(a\) and \(b\) feet (\(a>b\)); and if it is required to cut from it a beam of uniform square section; prove that the beam of greatest volume is \(\displaystyle\frac{al}{3(a-b)}\) feet long.
Prove that
Nine thin rods, freely jointed together, are arranged so as to form an equilateral triangle \(ABC\) together with the triangle \(DEF\) constituted by joining the middle points of the sides. This framework is symmetrically supported in a vertical plane by vertical strings attached to the joints at \(B\) and \(C\). Show that, if the weight of each of the three inner rods is \(W\) and the weight of each of the six outer rods is \(w\), then there is a tension in the rod \(EF\) of magnitude \[ \frac{2}{\sqrt{3}}(W+w). \]
A uniform rod rests with its ends on two smooth planes inclined at \(30^\circ\) and \(45^\circ\) respectively to the horizontal. Prove that the inclination of the rod will be \(\cot^{-1}(\sqrt{3}+1)\). Also find what weight, fixed to the rod at a quarter of its length from one end, will suffice to enable the rod to rest horizontally.
Three equal spheres are lying in contact on a horizontal plane and are held together by a string which passes round them. A tube of weight \(W\) is placed with one diagonal vertical so that its lower faces touch the spheres, and the cube is supported in this position by the spheres. Show that the tension in the string is \(\frac{1}{3\sqrt{6}}W\), all friction being neglected. Note: The scanned document contains `1/3 sqrt(6) W`, which is ambiguous. Transcribed as the most likely mathematical meaning.
Explain the difference between stable, unstable and neutral equilibrium. A heavy flexible chain of weight \(w\) per unit length hangs over a light circular pulley of radius \(a\), the free parts of the chain being of equal length. The pulley is mounted on a frictionless horizontal spindle. Show that the position is unstable, but that it will become stable, for a small displacement, if a weight \(W\) is attached to the lowest point of the pulley, provided that \[ W > 2wa. \]
A string of length \(2l\) and of uniform density \(w\) is fixed at \(A, B\), two points distant \(2a\) at the same level. A weight \(W\) is hung on the string at the mid-point \(C\). Find the equations of the two portions of the string.
A machine gun of mass \(M\) contains shot of mass \(M'\) and stands on a horizontal plane. Shot is fired at the rate \(m\) per second with velocity \(u\) relative to the ground. If the coefficient of sliding friction between the gun and the plane is \(\mu\), show that the velocity of the gun backwards by the time the mass \(M'\) is fired is \[ \frac{M'}{M}u - \frac{(M+M')^2-M^2}{2mM}\mu g. \]