Differentiate \(\cos^{-1}\frac{a+b\cos x}{b+a\cos x}\) and \(x^{1+x}\). If \(y=\sqrt{1-x^2}\sin^{-1}x\) shew that \((1-x^2)\frac{dy}{dx}+xy=1-x^2\) and obtain the expansion of \(y\) in a series of ascending powers of \(x\).
A segment is cut of the parabola \(y^2=4ax\) by a chord joining the points \((x_1, y_1)\) and \((x_2, y_2)\). Prove that the area is \((y_1 \sim y_2)^3/24a\), and that the coordinates of the centroid of the area are \(\frac{2y_1^2+y_1y_2+2y_2^2}{20a}\) and \(\frac{y_1+y_2}{2}\).
A flywheel weighing 40 lbs. has a radius of gyration 9 inches; it is driven by a couple fluctuating during each revolution so that the curve connecting couple and angular position during one revolution is a triangle, the couple becoming zero once per revolution and reaching a maximum of 2 ft. lbs. There is also a constant resisting couple such that the motion is the same in each revolution. If the flywheel makes 60 revolutions a minute, shew that the difference between the greatest and least angular velocities is approximately 5.7 per cent. of either.
A right circular conical tent has a given volume, find the ratio of its height to the radius of the base when as little canvas as possible is used. If the canvas is spread out what fraction does it form of a complete circle?
Two particles can move in the same straight line in a field of force per unit mass directed towards a point in that line and varying as the distance from that point. Shew that consecutive impacts between the particles take place at equal intervals of time and at one or other of two points in the line, and that the greatest distances between the particles during these intervals form a geometrical progression of ratio \(e\), where \(e\) is the coefficient of restitution between the particles.
Find \(\int \sin^{-1}x\,dx, \int\frac{\sin^2 x\,dx}{1+\cos^2x}, \int_0^\infty \frac{dx}{(1+x^2)^2}, \int_0^a \frac{x^3\,dx}{\sqrt{a-x}}\).
On a given day the depth at high water over a harbour bar is 32 ft., and at low water \(6\frac{1}{4}\) hours earlier it is 21 ft. If high water is due at 3.20 p.m., what is the earliest time at which a ship drawing 28 ft. 6 ins. can cross the bar, assuming the rise and fall of the tide to be simple harmonic?
Trace the curve \(r=a(2\cos\theta-1)\). Find the areas of the loops and shew that their sum is \(3\pi a^2\).
\(A, B, C, D\) are points in one plane. Forces are represented in magnitude and line of action by \(AB, BC, CA, DA, DB, DC\). Show that their resultant is a force represented by \(3DG\) in magnitude and direction, where \(G\) is the mean centre of \(A, B, C\) and that its line of action passes at a distance \(2\Delta/3DG\) from \(D\), where \(\Delta\) is the area of the triangle \(ABC\).
Obtain just enough conditions for the equilibrium of a system of forces in one plane. A bead of weight \(W\) rests on a smooth string attached to fixed points on the same level, the strings making angles \(\alpha\) with the horizontal. A horizontal force \(P\) is applied to the bead and a new equilibrium position assumed. Show that if the strings then make angles \(\theta, \phi\) with the horizontal, \(\theta\) and \(\phi\) are given by \[ P \sin\phi + W \cos\phi = W \cos\alpha = W \cos\theta - P \sin\theta. \]