Four equal rods each of length \(l\), freely jointed at their opposite corners, form a rhombus \(ABCD\). The opposite corners \(A, C\) are connected by an elastic string whose unstretched length is \(b (< 2l)\). The system is kept in equilibrium by forces at \(B\) and \(D\) acting inwards along the line \(BD\). Show that these forces have a maximum value when \[ \frac{AC}{BD} = \frac{\left(\frac{b}{2l}\right)^{\frac{1}{3}}}{\left\{1 - \left(\frac{b}{2l}\right)^{\frac{2}{3}}\right\}^{\frac{1}{2}}}. \]
A conic \(S\) is the polar reciprocal of itself with respect to another conic \(S'\). Prove that the conics touch at two distinct points \(P, Q\); that any chord of \(S\) through the pole of \(PQ\) is divided harmonically by \(S'\); and that \(S'\) is the polar reciprocal of itself with respect to \(S\).
Prove, by taking logarithms or otherwise, that if \(k, l, m, n, p, q, r\) are positive numbers of the form \(n-3, n-2, n-1, n, n+1, n+2, n+3\), the ratio of \(l^6 n^9 q^6\) to \(k m^7 p^{15} r\) is \(1 + 120n^{-6} + 1260n^{-8} + \dots\).
Discuss the connection between Newton's laws of motion and the fundamental statical postulates, such as the composition, independence and transmissibility of forces. Forces are represented in magnitude and line of action by coplanar lines \(AA', BB', \dots\). Prove that the resultant is represented in direction by \(GG'\) and in magnitude by \(n.GG'\), where \(n\) is the number of forces in the given system, and \(G\) is the centroid of the points \(A, B, \dots\), \(G'\) is the centroid of the points \(A', B', \dots\). Also, give a logical statement of the development from the fundamental postulates of the statical theorems on which you base your proof.
Prove that a rigid body possesses a centre of gravity such that if it be freely suspended at that point and allowed to hang under the influence of gravity it will be in equilibrium in any position. A tripod consists of three equal uniform rods \(AO, BO, CO\) rigidly connected at \(O\) so that they are at right angles to one another. If the tripod be hung from the point \(A\) show that the plane \(ABC\) makes an angle \(\tan^{-1} 2\sqrt{2}\) with the horizontal.
It is given that \[ k_1/(x-a_1) + k_2/(x-a_2) + \dots + k_n/(x-a_n) = 0, \] where \(k_1+k_2+\dots+k_n = 0\). Prove that, if \(x=(py+q)/(ry+s)\), \(a_1 = (pb_1+q)/(rb_1+s), \dots, a_n = (pb_n+q)/(rb_n+s)\), where \(ps-qr \neq 0\), then \[ k_1/(y-b_1) + \dots + k_n/(y-b_n) = 0. \]
Prove that \[ \frac{1}{1!(2n)!} + \frac{1}{2!(2n-1)!} + \frac{1}{3!(2n-2)!} + \dots + \frac{1}{n!(n+1)!} = \frac{2^{2n}-1}{(2n+1)!}. \]
Define the Potential Energy of a connected system of bodies under the action of given external forces. Give an outline of the theory by which, when the Potential Energy is known for all possible positions of the system, the positions of equilibrium and their stability can be investigated, stating the principles that are assumed in the investigation. Two equal smooth circular cylinders of radius \(c\) are fixed with their axes parallel and in the same horizontal plane at a distance \(b\) apart. A cube of side \(2a\) rests with two adjacent faces touching the cylinders. Shew that, if \(a+c<\sqrt{2}b\), and \(a^2+c^2>b^2\), there are two positions of equilibrium in which the plane through the highest and lowest edges of the cube makes an angle \(\cos^{-1}\{(a+c)/\sqrt{2}b\}\) with the vertical. Also shew that these positions are unstable.
Two stopping points of an electric tramcar are 440 yards apart. The maximum speed of the car is 20 miles per hour and it covers the distance between stops in 75 seconds. If both acceleration and retardation are uniform and the latter is twice as great as the former, find the value of each of them, and also how far the car runs at its maximum speed.
Find all the solutions of the simultaneous equations \[ \sin x = \sin 2y, \quad \sin y = \sin 2z, \quad \sin z = \sin 2x, \] each angle being restricted to be positive and less than \(\pi\).