Problems

A uniform cylinder rests inside a fixed hollow cylinder, whose axis is horizontal, and subtends an angle \(2\alpha\) at this axis. Another cylinder equal in every respect to the former is placed so as to rest in contact with both without disturbing the former, the axes of all three being parallel. Shew that, if all the surfaces are equally rough, the angle of friction must be greater than both \(\frac{\pi}{4}-\frac{\alpha}{2}\) and \(\alpha\).

A straight rod \(SH\), of length \(2c\), whose centre of gravity is at a distance \(d\) from its centre, is suspended by a fine weightless string, of length \(2c\sec\alpha\), tied to its two ends \(S\) and \(H\) and slung over a small smooth peg \(P\). Shew that, if \(dc\cos\alpha\)?

An aeroplane has a speed of \(u\) miles per hour and a range of action \(x\) miles out and \(x\) miles home in calm weather. Prove that in a steady wind of \(v\) miles per hour its range of action in a direction inclined at \(\theta\) to the wind is \[ \frac{x(u^2-v^2)}{u(u^2-v^2\sin^2\theta)^{\frac{1}{2}}}. \] Find also the magnitude and direction of the maximum range.

A particle is projected from a point on a smooth inclined plane. At the \(r\)th impact it strikes the plane at right angles, and at the \(n\)th impact it is at the point of projection; prove that the coefficient of restitution \(e\) is given by the equation \(e^n - 2e^r + 1 = 0\).

Two masses, each equal to \(m\) lb., are connected by a light spring which exerts a force \(\lambda\) poundals per foot of extension. They are placed on a rough horizontal plane, the coefficient of friction being constant and equal to \(\mu\). One of the masses is projected along the plane directly away from the other, and is travelling at \(v\) feet per second at the instant when the second mass begins to move. Shew that during the subsequent motion the maximum value of the tension in the spring is \[ mg\sqrt{\mu^2 + \frac{\lambda v^2}{2mg^2}} \text{ poundals.} \]

The wind resistance of a car weighing 2400 lb. is \(\frac{v^2}{20}\) lb. wt., when \(v\) feet per second is the speed relative to the air. The car is travelling at 60 feet per second on a straight level road with a following wind of 60 feet per second when the brakes are applied, producing, together with frictional resistance (other than wind resistance), a constant retarding force of 1125 lb. wt. Assuming that \(\log_e \frac{3}{2} = \frac{2}{5}\) and that \(\log_e \frac{5}{3} = \frac{1}{2}\), find how far the car travels before it stops.

(i) Find the asymptotes and points of inflexion of the curve \(y^2(x^2-1)=x^3\). Sketch the curve. (ii) Find the value of \((\cos x)^{1/x^2}\) as \(x\) tends to zero.

\(C\) is the centre of a circle of radius \(a\). \(P\) is a given point outside the circle. \(CP=c\), and \(CP\) cuts the circle in \(A\). Any line through \(P\) cuts the circle in \(Q\) and \(R\). If the angle \(CPQ\) is \(\theta\), prove that the area of the triangle \(QAR\) is a maximum when \(\sin\theta = \frac{a}{c\sqrt{2}}\).

Prove that

1. [(i)] \(\int_1^\infty \frac{dx}{x\sqrt{x^2-1}} = \frac{\pi}{2}\);
2. [(ii)] \(\int_0^\pi \frac{dx}{1+\cos\lambda\cos x} = \frac{\lambda}{\sin\lambda}\).

A uniform rod \(AB\) of length \(a\) and weight \(W\) is suspended in a horizontal position by two equal strings \(AD, BE\), of length \(l\), which are parallel and inclined at a given angle \(\alpha\) to the vertical, and by a string \(AF\) which passes to a point \(F\) in \(DE\) produced, so that \(EF=DE\). Find the tensions in the strings and prove that if \(l\) be such that the tension of \(AF\) is as small as possible, this tension will be \(W\sin\alpha\) or \(\frac{1}{2}W\sec\alpha\), according as \[ \alpha < \text{ or } > \frac{\pi}{4}. \]