A locomotive weighing 40 tons can pull 210 ten-ton trucks at 20 miles an hour on the level. The trucks will just run at 20 miles an hour down an incline of 1 in 320. How many trucks can the locomotive pull at that speed up the same incline? If the frictional resistance of the engine is 300 pounds weight, what is its horse-power?
Define the angular velocity of a lamina moving in any manner in its plane and shew how to determine it when the velocities of two points of the lamina are given. A circle \(A\) of radius \(a\) turns round its centre with angular velocity \(\omega\). A circle \(B\) of radius \(b\) rolls on the circle \(A\) and its angular velocity is \(\omega'\). Find the time taken
\(Q\) is the mean centre of the four points on a central conic the normals at which pass through \(P\). Prove that if \(PQ\) passes through the centre the conic is a rectangular hyperbola.
Establish a reduction formula for the integral \[ \int_0^\infty \frac{dx}{(1+x^2)^n},\] and hence evaluate \[ \int_0^\infty \frac{dx}{(1+x^2)^n}\] for any positive integral \(n\).
The position of a point moving in two dimensions is given in polar co-ordinates \(r, \theta\): find the component velocities and acceleration along and perpendicular to the radius vector. The velocities of a particle along and perpendicular to a radius vector from a fixed origin are \(\lambda r^2\) and \(\mu \theta^2\): find the polar equation of the path of the particle and also the component accelerations in terms of \(r\) and \(\theta\).
Define simple harmonic motion and establish its chief properties. A heavy particle hangs at one end of a light elastic string which is such that the period of a small vertical oscillation of the particle is \(2\pi T\). The string is moving vertically upwards with uniform velocity \(gT_o\), and the particle is in relative equilibrium. Shew that, if the upper end of the string is suddenly fixed, the string will become slack if \(T_o\) is greater than \(T\), and that in this case the new motion has a period \[ 2(\pi - \cos^{-1} T/T_o)T + 2(T_o^2 - T^2)^{\frac{1}{2}}.\]
A family of ellipses of the same eccentricity \(e\) have the origin as centre and pass through the point \((c, 0)\). Find the equation of the locus of their vertices.
Show that \[ \phi(x) = \frac{3 \int_0^x (1+\sec y)\log\sec y\,dy}{\{x+\log(\sec x+\tan x)\}\log\sec x}\] is an even function of \(x\), and that for small \(x\) we have approximately \[ \phi(x) = 1+\frac{1}{420}x^4.\]
\(n\) equal perfectly elastic spheres move with given velocities under no forces in the same straight line. Shew that the final velocities of the spheres depend only on their order and not on their initial distances apart.
Shew that \[ \frac{d^n}{dx^n} \left(\frac{1}{x^2+2x+2}\right) = (-1)^n n! \sin(n+1)\theta \sin^{n+1}\theta,\] where \[\cot\theta = x+1.\]