Define "specific resistance." Find the drop in volts per hundred yards of copper cable for a current density of 1000 amperes per sq. in., if the specific resistance of copper is \(0.66 \times 10^{-6}\) ohm when one inch is employed as the unit of length for measurement of the copper.
Shew, by use of the methods of the differential calculus, or otherwise, that \[ \frac{1}{2} < \frac{e^x}{e^x-1} - \frac{1}{x} < 1 \] for all positive values of \(x\).
A particle of mass \(m\) slides down the rough inclined face of a wedge of mass \(M\) and inclination \(\alpha\), which is free to move on a smooth horizontal plane. Shew that the time of describing any distance from rest is less than the time taken when the wedge is fixed in the ratio \(\{1 - \frac{m \cos\alpha(\cos\alpha+\mu\sin\alpha)}{M+m}\}^{\frac{1}{2}} : 1\).
Shew that the curve given by the equations \begin{align*} x &= at^2+2bt+c, \\ y &= a't^2+2b't+c' \end{align*} is a parabola and find the equation of the tangent at the point whose parameter is \(t\).
A battery of 5 ohms resistance is connected to a 20 ohm galvanometer and gives a deflection of 40 divisions. What will be the deflection when a 4 ohm shunt is put across the galvanometer terminals?
Shew that the function \[ -4c+4c^2+16c^3-16c^4, \] where \(c=\cos\theta\), has maximum values equal to 1 when \(\theta\) is equal to \[ -\frac{3\pi}{5}, -\frac{\pi}{5}, \frac{\pi}{5}, \text{ and } \frac{3\pi}{5}; \] and draw a graph of the function for values of \(\theta\) between \(-\pi\) and \(\pi\).
By proper choice of units the curve on a time base representing the acceleration of an electric train is a quadrant of a circle, whose centre is the origin. The initial acceleration is 2.5 ft. per sec. per sec., and the acceleration falls to zero in 20 seconds. Calculate the velocity acquired and the distance described in that time.
If the sides of a parallelogram are parallel to the lines \(ax^2+2hxy+by^2=0\) and one diagonal is parallel to \(lx+my=0\), shew that the other is parallel to \[ (hl-am)x+(bl-hm)y=0. \]
Prove that the pedal equation of an epicycloid or a hypocycloid, the origin being at the centre of the fixed circle, is \(r^2+mp^2=a^2\), where \(m\) is positive for a hypocycloid and between \(-1\) and \(0\) for an epicycloid. Shew that the pedal equation of the evolute is \[ r^2+mp^2 = (m+1)a^2. \]
A motor-car has its centre of gravity at a height \(h\) ft. midway between the axles, the wheel-base being \(l\) ft. Shew that the ratio between the least distances in which the car can be stopped by brakes acting, (a) on the front wheels, (b) on the back wheels, is \(\frac{1-\mu h/l}{1+\mu h/l}\), where \(\mu\) is the coefficient of adhesion between the tyres and the ground; the rotary inertia of the wheels may be neglected.