Prove that an inextensible string carrying a uniform load per unit horizontal length hangs in a parabola. A trolley-wire is carried on poles round a curve of 1200 feet radius. The poles are spaced 40 yards apart, and in the middle of each span the wire sags 6 inches below the points of support. If the wire weighs \(\frac{1}{2}\) lb. per foot, show that the resultant horizontal pull on each pole is very nearly 180 lb.
A smooth rod passes through a smooth ring at the focus of an ellipse whose major axis is horizontal, and rests with its lower end on the quadrant of the curve which is furthest removed from the focus. Find its position of equilibrium, and show that its length must be at least \[ \frac{3a}{4} + \frac{a}{4}\sqrt{1+8e^2}, \] where \(2a\) is the major axis and \(e\) is the eccentricity.
A mass \(M\) is drawn from rest up a smooth inclined plane of height \(h\) and length \(l\) by a string passing over a smooth pulley at the top of the plane and supporting a mass \(m\). Prove that \(M\) will just reach the top of the plane if the string breaks when \(m\) has fallen a distance \[ \frac{M+m}{m}\frac{hl}{h+l}. \]
The centre of gravity of a railway truck is situated midway between the axles and at a height of 3 feet above the rail level. The distance between the axles is 11 feet. When the truck is running at 30 miles per hour, the brakes are applied to the front wheels so as to lock them, the rear wheels remaining free. The coefficient of friction between the wheels and the rails being \(\frac{1}{5}\), prove that, neglecting the rotational inertia of the wheels, the truck will run \(126\frac{2}{3}\) yards before coming to rest.
A particle is projected along the inner surface of a smooth vertical sphere of radius \(a\), starting at the lowest point \(A\) with velocity \(\sqrt{\frac{7ag}{2}}\). Prove that the particle will leave the sphere at an angular distance of 60° from the top. Prove also that it will strike the sphere again at its lowest point \(A\).
A spring requires a force of \(P\) lb. weight to stretch it 1 inch. Find an expression for the potential energy stored in the spring when it is stretched \(x\) inches. A spring that requires a force of 18 lb. to stretch it 1 inch is placed on a smooth horizontal table, is fixed at one end, and is connected at the other to a mass of 12 lb. A bullet of mass half an ounce is fired along the axis of the spring with a velocity of 1540 feet per second, and embeds itself in the 12 lb. mass. Prove that the greatest extension of the spring in the subsequent motion is very nearly 2 inches.
Prove that \(y\) has a maximum value when \(\frac{dy}{dx}=0\) and \(\frac{d^2y}{dx^2}\) is negative. A Norman window consists of a rectangle surmounted by a semicircle. The perimeter of the window being given, prove that its area is a maximum when the radius of the semicircle is equal to the height of the rectangle.
Establish a formula for the \(n\)th differential coefficient of the product of two functions. Prove that \[ \frac{d^n}{dx^n}\left(\frac{\log x}{x}\right) = (-1)^n \frac{n!}{x^{n+1}}\left(\log x - 1 - \frac{1}{2} - \dots - \frac{1}{n}\right). \]
For a curve defined by the equation \(p=f(\psi)\) prove that the projection of the radius vector on the tangent is \(\frac{dp}{d\psi}\). For the curve \[ p = a \sin n\psi \] prove that \[ r^2 = a^2n^2 + (1-n^2)p^2. \]
Evaluate the integrals \[ \int \sqrt{a^2+x^2} \, dx, \quad \int \frac{dx}{(x-1)^{1/2}(x-2)}, \quad \int \left(\frac{1+x}{1-x}\right)^{1/2} dx. \]