10273 problems found
The mass of a train including the engine is 200 tons and the resistance to motion apart from brakes is 10 lb. weight per ton. The train starts from rest and travels 5 miles in 12 minutes ending at rest. The retardation is double the acceleration and both are uniform and there is a period during which the train runs at its maximum speed of 30 miles per hour. Find (i) the time of getting up full speed; (ii) the force exerted by the brakes; (iii) the rate at which the engine is working 1 minute from the start.
By induction or otherwise prove that \[ \frac{1}{n+1} - \frac{m}{n+2} + \frac{mC_2}{n+3} - \frac{mC_3}{n+4} \dots + \frac{(-1)^m}{m+n+1} = \frac{m!n!}{(m+n+1)!}, \] where \(m\) and \(n\) are positive integers.
A triangle with sides 5, 5, 6 has three circles inscribed in it each touching the other circles and two of the sides. Shew that the radii of the circles are 1, 1, and \(\frac{1}{2}(3-\sqrt{5})\).
Enunciate the principle of virtual work and explain how to apply it to find the position of equilibrium of a system having one degree of freedom when gravity is the only external force. A pentagon is formed of five equal uniform rods smoothly jointed at their extremities. It hangs with the two upper rods in contact with smooth pegs in the same horizontal line and the lowest rod horizontal. Shew that, if in equilibrium the pentagon is regular, the pegs must divide the rods in the ratio \(2+\sqrt{5}:3\).
Assuming that the resistance to the motion of a train is proportional to the square of the velocity and that if the engine exerts a constant pull the greatest velocity attainable is \(V\), shew that, if when the velocity attained is \(V_1 (
Assuming that the series \[ c(t) = 1 - \frac{t^2}{2!} + \frac{t^4}{4!} - \dots, \quad s(t) = t - \frac{t^3}{3!} + \frac{t^5}{5!} - \dots \] may be differentiated term by term, prove that the point with rectangular coordinates \(c(t), s(t)\) describes a circle with constant speed as the ``time'' \(t\) varies, and hence deduce that \[ c(t) = \cos t; \quad s(t) = \sin t. \]
Shew that if \(\alpha + \beta + \gamma = \frac{\pi}{4}\), then \[ (\sin\alpha + \cos\alpha)(\sin\beta + \cos\beta)(\sin\gamma + \cos\gamma) = 2(\sin\alpha\sin\beta\sin\gamma + \cos\alpha\cos\beta\cos\gamma). \]
Prove that the centre of gravity of the part of the surface of a sphere cut off by two parallel planes is midway between the planes. A thin hemispherical bowl of weight \(W\) contains a weight \(W'\) of water and rests on a rough inclined plane of inclination \(\alpha\). Shew that the plane of the top of the bowl makes an angle \(\phi\) with the horizontal given by \[ W \sin\phi = 2(W+W')\sin\alpha. \]
A shot is fired with velocity \(v\) ft. per sec. from the top of a cliff \(h\) ft. high and strikes a mark on the sea at a distance \(d\) ft. from the foot of the cliff. Find an equation to determine the direction of projection, and shew that the two possible directions of projection are at right angles if \(v^4h=gd^2\).
Show that, if \[ \frac{1}{1+u}e^{\frac{ux}{1+u}} = P_0(x) + P_1(x)\frac{u}{1!} + P_2(x)\frac{u^2}{2!} + \dots + P_n(x)\frac{u^n}{n!} \dots, \] then \[ P_n(x) = x^n - \frac{n^2}{1!}x^{n-1} + \frac{n^2(n-1)^2}{2!}x^{n-2} - \frac{n^2(n-1)^2(n-2)^2}{3!}x^{n-3} + \dots. \] By putting \(\frac{u}{1+u}=t\), deduce that \[ x^n = P_n(x) + \frac{n^2}{1!}P_{n-1}(x) + \frac{n^2(n-1)^2}{2!}P_{n-2}(x) + \dots. \]