Problems

(i) If \(e^y+e^{-x}=2\), prove that \[ \frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + \frac{dy}{dx} = 0. \] (ii) If the equiangular spiral \(r=ae^{\theta\cot\alpha}\) cuts any straight line through the origin in the consecutive points \(P_1, P_2, \dots, P_n\), and if \(\rho_n\) is the radius of curvature at \(P_n\), prove that \[ \log \frac{\rho_n}{\rho_m} = 2\pi(n-m)\cot\alpha. \]

Within a given circle of radius \(r\) an ellipse is drawn having double contact with the circle, and having one end of its minor axis at the centre of the circle. Prove that the maximum area the ellipse can have is \[ \frac{2\pi r^2}{3\sqrt{3}}. \]

(i) Prove that \[ \int_1^\infty \frac{dx}{x(1+x^3)} = \frac{2}{3}\log_e 2. \] (ii) Find the area of the curve \[ a^2y^2 = x^2(a^2-x^2). \]

A see-saw consists of a plank of weight \(w\) laid across a fixed rough log whose shape is a horizontal circular cylinder. The inclination to the horizontal at which it balances is increased to \(\alpha\) when loads \(W, W'\) are placed at the lower and higher ends respectively: and the inclination is reduced to \(\beta\) when the loads are interchanged. Show that the inclination of the plank when unloaded is \[ \frac{w'(W+W'+w)(W'\alpha - W\beta)}{w(W+W'-w')(W-W')}, \] \(w'\) being the load which, placed at the higher end, would balance the plank horizontal.

Explain the Principle of Virtual Work. A smooth sphere of radius \(r\) and weight \(W\) rests in a horizontal circular hole of radius \(a\). A string is wrapped once round the sphere above the hole and then pulled tight. What tension in the string will just raise the sphere?

A number of small rings can slide freely on a smooth fixed circular wire, and each ring repels every other ring with a force which is measured by the product of their masses and the distance between them: show that, in equilibrium, the centre of gravity of the rings is the centre of the circle.

A uniform cubical block of edge \(l\) is placed on the top of a fixed perfectly rough sphere, the centre of a face of the block being in contact with the highest point of the sphere. Show that the least radius of the sphere for which the equilibrium is stable is \(\frac{1}{2}l\).

A uniform chain of length \(2l\) is hung between two points at the same level distant \(2b\) apart. Find the equation of the curve, and the vertical and horizontal components of the tension at any point.

A mass \(M\) rests on a smooth table and is attached by two inelastic strings to masses \(m, m'\), (\(m' > m\)), which hang over smooth pulleys at opposite edges of the table. The mass \(m'\) falls a distance \(x\) from rest, and then comes into contact with the floor (supposed inelastic). Show that \(m\) will continue to ascend through a distance \(y\) given by \[ y = \frac{(m'-m)(M+m)}{m(M+m+m')}x. \] Show further that when \(m'\) is jerked into motion again as \(m\) falls it will ascend a distance \[ \frac{(M+m)^2}{(M+m+m')^2}x. \]

The external resistance to the motion of a bicycle consists of a constant force together with a force proportional to the square of the velocity. The speed is constant at 10 ft. sec. when free wheeling down a slope of 1 in 50, and constant at 20 ft. sec. on a slope of 1 in 25. The mass of the bicycle and the rider is 200 lbs. Find the power expended in maintaining a steady speed of 15 ft. sec. on the level, assuming that 15 per cent. of the work is lost in friction in the pedalling gear.