Prove that \begin{align*} &\cos^2 x \cos (y + z - x) + \cos^2 y \cos (z + x - y) + \cos^2 z \cos (x + y - z) \\ &= 2 \cos x \cos y \cos z + \cos (y + z - x) \cos (z + x - y) \cos (x + y - z). \end{align*}
The curve connecting velocity and time for a moving body is a symmetrical arc of a circle 4 in. long and 1 in. high at the centre. The body starts from rest and comes to rest again at the end. The vertical scale is 1 in. = 20 ft. per sec., and the horizontal scale is 1 in. = 10 sec. Find the maximum acceleration in ft. per sec. per sec., and the distance described in feet.
Prove that \(\displaystyle\frac{\cot 3x}{\cot x}\) never lies between 3 and \(\frac{1}{3}\).
Explain and contrast the nature and laws of sliding and rolling friction. A light string, supporting two weights \(w\) and \(w'\), is placed over a wheel (radius \(a\)) which can turn round a fixed rough axle (radius \(b\), friction coefficient \(\mu\)). There being no slipping of the string on the wheel, shew that the wheel will just begin to rotate round the axle if \((w-w')a = (w+w'+W)b\sin\epsilon\) where \(\mu=\tan\epsilon\) and \(W\) is the weight of the wheel. A ladder is placed in a vertical plane with one end on a rough horizontal floor (\(\mu\)) and the other against a rough vertical wall (\(\mu'\)). (i) Find the inclinations for which equilibrium is possible. (ii) When the inclination and friction coefficients are such that equilibrium is not possible, shew how to determine the position and weight of the smallest mass which when suspended from a rung of the ladder will produce equilibrium.
Explain briefly the principle of virtual work. A frame to form a girder consists of 19 rods of equal length hinged together to make nine equilateral triangles, the lower boom of the girder consisting of five horizontal rods in tension, and the upper boom consisting of four horizontal rods in compression. The ends of the girder rest on fixed supports \(A\) and \(B\), and the only load is at the lower joint whose distance from \(A\) is \(\frac{2}{5}\) of the span. Shew by the principle of virtual work that if each of the two horizontal rods next to the load is stretched 0.05 inches, the other rods remaining unchanged, the consequent drop of the load is \(\frac{19\sqrt{3}}{300}\) inches.
Prove that, if \(n\) angles of which no two differ by a multiple of \(\pi\) satisfy the relation \[ p_0 + p_1 \cot\theta + p_2 \cot^2\theta + p_3 \cot^3\theta + \dots p_n \cot^n\theta = 0, \] the cotangent of the sum of these angles is \[ - (p_0 - p_2 + p_4 - p_6 + \dots) \div (p_1 - p_3 + p_5 - p_7 + \dots). \] Hence or otherwise prove that the relation \[ \cot\theta = \frac{a + a_1 \operatorname{cosec}^2\theta + a_2 \operatorname{cosec}^4\theta + \dots a_r \operatorname{cosec}^{2r}\theta}{b + b_1 \operatorname{cosec}^2\theta + b_2 \operatorname{cosec}^4\theta + \dots b_s \operatorname{cosec}^{2s}\theta} \] is generally satisfied by either \(2r\) or \(2s+1\) values of \(\cot\theta\), whichever of these numbers is the greater: and that, if all these values are real, the cotangent of the sum of the corresponding angles is \(a/b\).
A fleet is steaming due N. at 10 knots, and a cruiser which can steam 18 knots is ordered to proceed at full speed on a N.E. course for 5 hours: she is then to rejoin the fleet as quickly as possible. What course should she then steer and when will she rejoin?
An observer sees an aeroplane due N. at an elevation of \(10\frac{1}{4}^{\circ}\). Two minutes later he sees it N.E. at the same angular elevation. It is known to be going due E. at a speed of 60 miles per hour. Show that it is rising at a rate of 412 feet per minute.
Deduce the equations of equilibrium for a uniform freely suspended string, shewing that the string hangs in the form of a catenary. Prove that the vertical tension at any point is equal to \(w\lambda\), where \(w\) is the weight of unit length of string, and \(\lambda\) is the vertical distance from the given point to the directrix of the catenary. A heavy string of length \(2l\) is hung from two fixed points \(A, B\) in the same horizontal line, at a distance apart equal to \(2a\). A weight \(W\) is attached to a certain point of the string. Shew that the parameters of the two catenaries in which the string hangs are the same, and shew that if \(W\) is in the middle of the string and its weight is great in comparison to that of the string, the parameter \(c\) is equal to \(Wa/2w\sqrt{l^2-a^2}\) nearly, while if on the other hand the weight of \(W\) is small in comparison with that of the string, \[ l=c\sinh\frac{a}{c} + \frac{W}{2w}\left\{\cosh\frac{a}{c}-1\right\}. \]
The springs of a motor car are such that the weight of the parts carried on the springs depresses the latter through 2 inches from the position when unloaded. Find the natural period in which the car bounces; and shew that, on a road in which there is a series of ridges at intervals of 6 ft., the bouncing may become excessive at a speed of about 9 miles an hour.