Any number of wrenches, all of the same pitch, have as axes generators of the same system of a hyperboloid of one sheet; shew that the wrenches are in complete equilibrium if a concurrent system of forces equal and parallel to their forces be in equilibrium.
A particle of mass \(m\) impinges at right angles with velocity \(V\) upon a uniform rod of mass \(M\) and length \(2a\). If the rod is initially at rest and is struck at a distance \(b\) from its centre, prove that the loss of kinetic energy due to the impact is \[ \frac{1}{2}(1-e^2)\frac{Mma^2}{Ma^2+m(a^2+3b^2)}V^2, \] where \(e\) is the coefficient of elasticity.
A triangle is immersed in water, and its corners are at depths \(\alpha, \beta, \gamma\) below the surface, shew that the depth of its centre of pressure is \[ \frac{\alpha^2+\beta^2+\gamma^2+\beta\gamma+\gamma\alpha+\alpha\beta}{2(\alpha+\beta+\gamma)}, \] neglecting the atmospheric pressure. A trapezium is immersed; \(\alpha, \beta\) are the depths of the corners at the ends of one of its non-parallel sides, \(\gamma, \delta\) the depths of the other corners; shew that the depth of its centre of pressure is \[ \frac{1}{2}\frac{(\alpha+\beta)(\alpha^2+\beta^2)-(\gamma+\delta)(\gamma^2+\delta^2)}{\alpha^2+\alpha\beta+\beta^2-\gamma^2-\gamma\delta-\delta^2}. \]
A coaxial system of thin convergent lenses, of numerical focal lengths \(f_1, f_2 \dots f_n\), is such that the distance between any adjacent pair is twice the sum of their focal lengths. Shew that (i) the linear magnification for a small object at distance \(2f_1\) in front of the first lens is \((-1)^n\); (ii) that the first principal focus of the entire system is at a distance \([\Sigma(1/f)]^{-1}\) from this point measured towards the system. Determine the power of the system, and the positions of its points of unit magnification, as \(n\) is odd or even.
A solid sphere of radius \(a\) and density \(\sigma\) is surrounded by liquid of density \(\rho_1\) enclosed within a massless concentric shell of inner radius \(b\) and outer radius \(c\); the space outside extending to infinity is occupied by liquid of density \(\rho_2\). The sphere is set in motion with velocity \(U\). Shew that at any point on either the inner or outer surface of the shell, which is at angular distance \(\theta\) from the direction of \(U\), the initial impulsive pressure is \[ \frac{3}{4}a^3b^3cU\cos\theta\rho_1\rho_2/\{p_1b^3(2b^3+a^3)+p_2c^3(b^3-a^3)\}; \] and that the impulse to be applied to the sphere is \[ \frac{4\pi}{3}a^3U\left\{\sigma+\frac{1}{2}\rho_1\frac{2\rho_1 b^3(b^3-a^3)+\rho_2 c^3(b^3+2a^3)}{\rho_1 b^3(2b^3+a^3)+\rho_2 c^3(b^3-a^3)}\right\}. \]
Any point \(S\) on a sphere is displaced on the great circle through a fixed point \(O\) on the sphere to a point \(S'\) by the aberration law \(\sin SS' = k\sin SO\), where \(k\) is any finite number less than unity. Prove that, if \(S\) lie on a great circle having any fixed point \(P\) as pole, \(S'\) lies on a small circle with \(P\) as pole.