Problems

\(A\) and \(B\) are two points at the same level, and \(4a\) apart. \(AC, BD\) are two equal uniform rods of length \(a\sqrt{2}\), free to turn about \(A\) and \(B\). \(C\) and \(D\) are \(2a\) apart, and at a depth \(a\) below \(AB\), being joined by a uniform chain of weight \(W\), which rests in equilibrium with its middle point at a small depth \(\frac{1}{16}a\) below \(CD\). Prove that the weight of each rod is \(7W\) approximately.

Two smooth and perfectly elastic spheres of equal radii, but of masses 1 lb. and 4 lb. respectively, are at rest on a smooth horizontal table. The heavier sphere is projected horizontally and strikes the lighter sphere. Prove that the greatest angle through which the direction of motion of the heavier sphere can be deflected by the impact is \(\sin^{-1}\frac{1}{4}\).

A rigid body consisting of two equal masses joined by a weightless rod rests on a smooth horizontal table. One of the masses receives a horizontal blow perpendicular to the rod. Prove that each mass describes a cycloid. \par If the body is thrown up in the air in any manner, and air resistance is neglected, describe the motion in general terms.

A uniform heavy flexible rope \(AOB\) hangs over a small fixed peg \(O\). The lengths \(OA, OB\) hanging freely on either side are \(2a\) and \(a\) respectively, and the rope is in limiting equilibrium, on the point of slipping round the peg. If \(OA\) is now slightly increased, so that slipping begins, prove that the velocity \(v\) with which the end \(B\) reaches the peg is given by \[ v^2 = 6ga(4\log_e \frac{4}{3}-1). \]

A ship of mass 5000 tons is coming to rest with engines stopped. The resistance to motion is \(cv+ev^2\) tons weight, where \(v\) is the speed of the ship in ft. per sec. and \(c\) and \(e\) are constants. In 1000 feet run the speed falls from 20 ft. per sec. to 12 ft. per sec., and in the next 1000 feet it falls to 6 ft. per sec. Prove that \(c=0 \cdot 54\) and \(e=0 \cdot 045\), and find the speed of the ship after another 1000 feet run. \par [Assume that \(g=32\) ft. per sec. per sec., and that \(\log_e \frac{4}{3} = 0 \cdot 288\).]

(i) Prove that \(\frac{x}{a}+\frac{y}{b}=1\) touches the curve \(y=be^{x/a}\) at the point where the curve crosses the axis of \(y\). \par (ii) If \(x = y\sqrt{y^2+2}\), prove that \[ (1+x^2)\frac{d^2y}{dx^2}+x\frac{dy}{dx}=\frac{1}{4}y. \]

\(AA'\) is the major axis of an ellipse. Any line through \(A\) cuts the ellipse in \(P\), and the circle on \(AA'\) as diameter in \(Q\). The angle \(PAA' = \theta\), and \(e\) is the eccentricity of the ellipse. Prove that the length of \(PQ\) is a maximum when \[ 2e^2\cos^2\theta = 3-e^2 - \sqrt{(1-e^2)(9-e^2)}. \]

Prove that

1. [(i)] \(\int_0^3 \frac{x\,dx}{\sqrt{3+6x-x^2}} = \pi - \sqrt{3}\).
2. [(ii)] \(\int_0^{\pi/3} (1+\tan^6\theta)\,d\theta = \frac{8}{5}\sqrt{3}\).
3. [(iii)] \(\int_1^3 \frac{x^2+1}{x^4+7x^2+1}\,dx = \frac{1}{3}\tan^{-1}\frac{1}{2}\).

A Venetian blind is 7 feet long when fully stretched out, and 1 foot long when completely drawn up. There are 30 movable strips, and each weighs one pound. Find the work done in raising the blind against gravity.

A rectangular picture frame hangs from a smooth peg by a string of length \(2a\) whose ends are attached to points on the upper edge at distances \(c\) from the middle point. Shew that if the depth of the picture exceeds \(\frac{2c^2}{\sqrt{a^2-c^2}}\), the symmetrical position is the only position of equilibrium. \par If the depth \(2d\) of the picture is less than the critical value given above, find the inclination of the picture to the horizontal in the other equilibrium position.