A Wheatstone bridge has resistances as shown and \(A,B\) are maintained at a constant potential difference \(E\). Show that when \(S\) is large compared with \(R_1\) and \(R_2\), the current through the galvanometer is approximately \[ E(R_2-R_1)/S(R_2+R_1), \] and that the potential difference between \(A\) and \(C\) is approximately \[ ER_1/(R_1+R_2). \] The wires are uniform and made of the same material of specific heat \(\sigma\), and their resistances increase with the temperature \(\theta\) according to the law \[ R_1=r_1(1+\alpha\theta), \quad R_2=r_2(1+\alpha\theta), \] where \(\alpha\) is small. Under the influence of the currents each wire is supposed to heat up uniformly from the temperature \(\theta=0\) without loss of heat. Show that, to the first order in \(\alpha\), \(R_1\) and \(R_2\) increase linearly with the time. If the bridge is initially in balance, show that at a time \(t\) after the potential difference between \(A\) and \(B\) has been applied the current through the galvanometer is approximately \[ \frac{aE^3t}{SJ\sigma(r_1+r_2)^3}\left(\frac{r_2^2}{m_2}-\frac{r_1^2}{m_1}\right), \] where \(J\) is the mechanical equivalent of heat, \(m_1\) the mass of each wire \(AC,DB\) and \(m_2\) the mass of each wire \(AD,CB\).
Two conics touch at \(A\) and intersect in \(B\) and \(C\). A line through \(A\) meets the conics in \(P\) and \(Q\). Shew that the tangents at \(P\) and \(Q\) meet on \(BC\). State the reciprocal theorem.
Shew that there are two spheres, real, coincident, or imaginary, which pass through three given points and touch a given plane, explaining how the three cases arise. Find also how many spheres pass through three given points and touch a given line.
Shew that, if \(\tan\alpha, \tan\beta, \tan\gamma\) are all different and such that \[ \tan 3\alpha = \tan 3\beta = \tan 3\gamma, \] then \[ (\tan\alpha+\tan\beta+\tan\gamma)(\cot\alpha+\cot\beta+\cot\gamma) = 9. \] If \(\tan\alpha:\tan\beta:\tan\gamma = a:b:c\), shew that \(\tan^2\alpha = \displaystyle\sqrt{-\frac{3a^2}{bc+ca+ab}}\). Generalize the first theorem to the case of \(2n+1\) angles \(\alpha, \beta, \dots, \lambda\).
Give without proof expressions for \(\sin\theta, \cos\theta\) in terms of \(t \left(=\tan\frac{\theta}{2}\right)\). If \(\theta\) is an acute angle, shew that \[ \frac{\tan\theta}{\theta} > \frac{\sin\theta}{\theta}. \] Hence, or otherwise, prove that the equation \[ \frac{1}{\sin\theta} - \frac{1}{\theta} = k \] is satisfied by one and only one acute angle \(\theta\) if \(0 < k < 1 - \frac{2}{\pi}\), and by no acute angle \(\theta\) if \(k\) lies outside these limits.
If \(ax+by+cz=1\) and \(a,b,c\) are positive, shew that the values of \(x,y,z\) for which \(\displaystyle\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\) is stationary are given by \[ ax^2=by^2=cz^2. \] Shew that this is a true maximum or minimum if \(xyz > 0\).
Define the mean value of \(f(x)\) with respect to \(x\) for values of \(x\) lying in an interval \((a,b)\). A point moves along a straight line in such a way that \[ v_t = v_s+ks, \] where \(v_t, v_s\) are the mean values of the velocity with respect to the distance travelled \(s\) and the time taken \(t\) respectively, and \(k\) is a constant. Shew that \(s,t\) satisfy the equation \[ \frac{ds}{s} = \frac{dt}{t}\{1+kt \pm \sqrt{kt(2+kt)}\}. \] Interpret this solution in the case \(k=0\), and shew on general grounds that a negative value of \(k\) is inadmissible.
A uniform thin hollow right circular cylinder stands upright on a table, and three smooth equal spheres each of weight \(w\) are placed inside it. The ratio of the radius of a sphere to that of the cylinder is \(\alpha\). Prove that if \(\frac{1}{2} > \alpha > 2\sqrt{3}-3\), so that two of the spheres rest upon the ground, the cylinder will not overturn if its weight exceed \(\frac{w}{2}(1+\sqrt{1-2\alpha^2})\). Each sphere is to be taken in contact with the cylinder and with the other two spheres.
Two small rings \(P, Q\) can slide on the upper part of a smooth circular wire in a vertical plane, and are attached by strings of equal length to a third ring \(R\) which is free to slide along the vertical diameter of the circle. The weights of the three rings are equal. Prove that, if the lengths of the strings are less than the radius of the circle, there is a stable position of equilibrium in which \(R\) is at the centroid of the triangle \(POQ\), where \(O\) is the centre of the circle.
A bucket of mass \(m_1\) is joined to a counterpoise of mass \(m_2\) by a light string hanging over a smooth pulley. A ball of mass \(m\) is dropped into the bucket. Shew that the ball will come to rest in the bucket at a time \(\displaystyle\frac{ev(m_1+m_2)}{(1-e)m_2g}\) after the first impact, where \(v\) is the velocity of the ball relative to the bucket immediately before the first impact, and \(e\) is the coefficient of restitution. Shew that the sum of the upward momentum of the system on one side of the pulley and the downward momentum of that on the other side, increases at a uniform rate, and determine this rate. Hence or otherwise shew that the velocity of the system so soon as the ball has come to rest in the bucket is \[ u + \frac{mm_2+e(mm_1+m_1^2-m_2^2)}{(1-e)m_2(m+m_1+m_2)}v, \] where \(u\) is the downward velocity of the bucket immediately before the first impact.