Problems

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.

Forces \(X, Y, Z\) act along the sides \(BC, CA, AB\) of a triangle \(ABC\) (supposed not equilateral), and are such that the resultant is a force of magnitude \(P\) in the line joining the circumcentre \(O\) to the incentre \(I\) of the triangle \(ABC\). Prove that \[ \frac{X}{\cos B - \cos C} = \frac{Y}{\cos C - \cos A} = \frac{Z}{\cos A - \cos B} = \frac{R}{4OI\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}}P, \] where \(R\) is the circum-radius.

A circular cylinder of radius \(a\) and weight \(W\) rests with its axis horizontal in a V-shaped groove whose sides are inclined at equal angles \(\alpha\) to the horizontal, and a gradually increasing couple is applied to it in a plane perpendicular to its axis. If \(\alpha<\lambda\), where \(\lambda\) is the angle of friction, prove that the cylinder will begin to roll up one side of the groove when the couple exceeds \(Wa\sin\alpha\). If \(\alpha>\lambda\), prove that the cylinder will begin to rotate about its axis when the couple exceeds \[ \frac{1}{2}Wa\sec\alpha\sin 2\lambda. \]

An aeroplane has a speed \(u\), and a range of action \(R\) (out and home) in calm weather. If there is a north wind of velocity \(v\), prove that the range of action in a direction making an angle \(\phi\) with the north-south line is \[ R \frac{u^2-v^2}{u\sqrt{u^2-v^2\sin^2\phi}}, \] if \(vu\)? In this case sketch the locus of the extreme distances which can be reached in a non-stop flight out and home.