Three equal uniform rods PA, PB, PC, each of length \(2l\) and weight \(W\), are freely jointed at P and form a tripod with the feet A, B, C on a smooth horizontal plane. At three points D, E, F on the rods, such that AD=BE=CF=\(a\), are attached strings of equal length \(b\), and the other ends of the strings are fastened to a weight \(W'\). Assuming that the tripod is in equilibrium, with \(W'\) above the plane, shew that the angle (\(\theta\)) made by the rods with the vertical, and the angle (\(\phi\)) made by the strings with the vertical are given by the relations \[ \frac{\tan\theta}{\tan\phi} = \frac{W'(2l-a)}{3Wl+W'a}, \quad \frac{\sin\theta}{\sin\phi} = \frac{b}{2l-a}. \]
A submarine observes an approaching cruiser, steaming with velocity \(u\); the distance from the cruiser to the submarine is \(a\) and makes an acute angle \(\theta\) with the cruiser's course. The speed of the submarine is \(v\) and is less than \(u\); find a construction for the direction in which the submarine should steer in order to close as quickly as possible, and shew that, if \(\sin\theta > v/u\), the submarine can never get closer to the cruiser than the distance \[ \frac{a}{u}\{\sqrt{(u^2-v^2)}\sin\theta - v\cos\theta\}. \]
An electric train starts with an acceleration \(f\): but the acceleration diminishes uniformly with the time, becoming zero after a time \(t_0\) has elapsed since the start. Shew that in a time \(t\) (less than \(t_0\)), the space described is equal to \[ \frac{1}{6}ft^2(3t_0-t)/t_0. \] After the time \(t_0\) has elapsed, the acceleration remains zero; find a formula for the whole space described, when \(t\) is greater than \(t_0\).
A ship of mass 8000 tons slows, with engines stopped, from 12 knots to 6 knots in a distance of 1500 feet; calculate (in tons weight) the average force of resistance to the ship in slowing down. Assuming that the same resistance is experienced in increasing speed, calculate the horse-power necessary in order that the ship may regain its original speed in a distance of 2000 feet. [A knot is a speed of 100 feet per minute.]
Particles are projected from a point P with velocity \(\sqrt{(2gh)}\) in all directions in a vertical plane through P: shew that the parabolic paths lie within a parabola, having its focus at P and its latus-rectum equal to \(4h\). If P is at the top of a hemispherical mound of radius \(r\), prove that no particle can fall clear of the mound unless \(r < 4h\).
A load \(W\) is to be raised by a rope, from rest to rest, through a height \(h\): the greatest tension which the rope can safely bear is equal to \(nW\). Shew that the least time occupied in the ascent is equal to \[ \sqrt{\left(\frac{2nh}{(n-1)g}\right)}. \]
A railway truck of mass 12 tons moving at a speed of 5 feet per second runs into a stationary truck of mass 8 tons: after the collision the buffer springs start to be compressed. Calculate the kinetic energy (in foot-tons) of the two trucks, when the compression of the springs is a maximum. If each of the four buffer springs affected by the collision is such that a force of 2 tons weight is required to hold it compressed through a distance of 4 inches, calculate the maximum reaction during the collision, and the time of collision.
Prove that, if \begin{align*} ax+by+cz &= 0, \\ ax^2+by^2+cz^2 &= 0, \end{align*} and then \[ ax^3+by^3+cz^3 = (a+b+c)xyz. \]
A man has 4 shillings and 6 pennies, and wishes to give each of six boys a shilling, a penny, or a shilling and a penny; shew that he can do so in 473 different ways.
Having given \[ \sin\alpha=a, \sin\beta=b, \sin\gamma=c, \sin\delta=d, \] and \(\alpha+\beta+\gamma+\delta=\pi\), prove that \(\Sigma a^4 - 2\Sigma a^2b^2 + 4\Sigma a^2b^2c^2 + 4abcd(\Sigma a^2 - 2)=0\). Hence, or otherwise, shew that if a quadrilateral inscribed in a circle has its sides proportional to 1, 2, 3, 4, the length of the longest side is \(\sqrt{\frac{38}{55}}\) of the diameter.