Prove that the length of the subnormal of the curve \[ y=f(x) \text{ is } y\frac{dy}{dx}. \] In the catenary \[ y=c\cosh\frac{x}{c}; \] prove that the subtangent is \(c\coth\frac{x}{c}\), the subnormal is \(\frac{1}{2}c\sinh\frac{2x}{c}\), and the normal is \(\frac{y^2}{c}\).
If \(y=a+x\sin y\), where \(a\) is a constant, prove that, when \(x=0\), \[ \frac{dy}{dx} = \sin a, \text{ and } \frac{d^2y}{dx^2} = \sin 2a. \] Hence by Maclaurin's Theorem expand \(y\) in powers of \(x\) as far as \(x^2\).
If forces are represented in magnitude and direction by \(\lambda \cdot OA, \mu \cdot OB, \nu \cdot OC, \dots\), prove that their resultant is represented in magnitude and direction by \((\lambda+\mu+\nu+\dots)OG\), where \(G\) is the centre of gravity of masses \(\lambda, \mu, \nu, \dots\) at \(A, B, C, \dots\). \(P, Q, R\) are taken on the sides \(BC, CA, AB\) of a triangle dividing each in the same ratio \(1+\lambda:1-\lambda\) in the same sense round the triangle. Prove that forces represented by \(AP, BQ, CR\) are equivalent to a couple whose moment is \(2\lambda\Delta\), where \(\Delta\) is the area of the triangle.
A rectangle is hung from a smooth peg by a string of length \(2a\) whose ends are fastened to two points on the upper edge at distances \(c\) from the middle point. Show that an oblique position of equilibrium is possible if the depth of the rectangle is less than \(\displaystyle\frac{2c^2}{\sqrt{a^2-c^2}}\).
\(ABCD\) is a rhombus of freely jointed rods in a vertical plane and \(B, D\) are connected by a rod jointed to the rhombus. \(A\) and \(B\) are fixed so that \(AB\) is horizontal and below the level of \(CD\). The acute angle \(A\) of the rhombus is \(\alpha\). If a weight \(W\) is hung from \(C\), draw the force diagram and find the stress in the rod \(BD\) in terms of \(W, \alpha\).
A railway truck is at rest on an incline of slope \(\alpha\) with the lower pair of wheels locked. Show that the coefficient of friction \(\mu\) between the wheels and the rails must not be less than \((a+b)/(h+b\cot\alpha)\), where \(h\) is the distance of the centre of gravity of the truck from the plane of the rails, and \(a,b\) are the distances of the centre of gravity from the lower and upper axles measured parallel to the incline.
Explain how the potential energy of a system determines the equilibrium positions of a system and their stability. \(AB\) is the horizontal diameter of a circular wire whose plane is vertical. A bead of mass \(M\) at the lowest point \(C\) can slide on the wire and is attached to two strings which pass through small fixed rings at \(A, B\). To the other ends of the strings are attached equal masses \(m\) which hang freely. Find the potential energy of the system when it is displaced so that the radius to the bead makes an angle \(\theta\) with the vertical. Show that the equilibrium with \(M\) at \(C\) is stable if \(m < M\sqrt{2}\).
Two smooth elastic balls collide with given velocities in given directions; find the transference of momentum. If the balls approach along parallel lines with equal but opposite momenta, show that after oblique impact they move along parallel lines which are further apart than the first pair of lines.
A particle is projected under gravity with velocity \(\sqrt{2ga}\) from a point at a height \(h\) above a level plain. Show that the angle of elevation \(\theta\) for maximum range on the plain is given by \(\tan^2\theta = \displaystyle\frac{a}{a+h}\), and that the maximum range is \(2\sqrt{a(a+h)}\).
Find the horse-power required to lift 1000 gallons of water per minute from a canal 20 feet below and project it from a nozzle whose cross-section is 2 square inches, given that a cubic foot of water weighs \(62\frac{1}{2}\) lbs. and that 1 gallon of water weighs 10 lbs.