Prove the relation in isotropic material between Young's modulus \(E\), the modulus of rigidity \(C\) and Poisson's ratio \(\frac{1}{m}\), \(E = 2C\left(1 + \frac{1}{m}\right)\). A bar is subject to a normal stress \(p\), uniform over the cross-section. Show that the strain in a line in the bar making an angle \(\theta\) with the direction of the stress \(p\) is \[\frac{p}{E}\left(1 - \frac{2}{m}\sin^2\theta\right).\]
Prove that if \[ax^2+2hxy+by^2+2gx+2fy+c=0,\] then \[\frac{d^2y}{dx^2} = \frac{abc+2fgh-af^2-bg^2-ch^2}{(hx+by+f)^3}.\]
The weight of a car is 3200 pounds and the resistance to its motion consists of a constant frictional resistance of 34 pounds weight and a wind resistance proportional to the square of its speed. If the engine exerts a constant pull of 100 pounds weight and the maximum speed attainable is 45 miles per hour, determine its acceleration at 30 miles an hour, and shew that the distance traversed in speeding up from 15 to 30 miles an hour is 1551 feet, given \(\log_e 2 = \cdot 69315\), \(\log_e 10 = 2\cdot 30258\).
An equilateral triangle has its centre at the origin and one of its sides is \(x+y=1\), find the equations of the other sides. Prove that \(x^3+3x^2y-3xy^2-y^3=0\) represents the perpendiculars from the vertices of the triangle on the opposite sides.
A pair of conductors are laid side by side, and each one forms a closed curve. Each is of length 3000 yards and has a resistance of 0.05 ohm per 1000 yards. At one point the conductors are kept at a difference of pressure of 200 volts, and current is taken off between them at distances reckoned from one side of this point as follows: 200 amperes at 500 yards, 200 at 1500 yards, and 100 at 2000 yards. Find the current in each section of the ring and the difference of pressure at each of the points at which the current is taken off.
Trace the curve \[r^3\sin 4\theta = \sin(\theta+\alpha)\] (a) when \(0 < \alpha < \frac{1}{4}\pi\) and (b) when \(\alpha=0\). Shew how the shape of the curve passes over into its limiting form as \(\alpha\) tends to zero.
A train weighing 280 tons is drawn from rest up an incline of 1 in 140 against a frictional resistance of 16 pounds weight per ton. If the total pull of the engine varies with the distance according to the following table, find the velocity of the train after passing over 1000 feet.
Conics are drawn with \(A, A'\) as the extremities of a principal axis; prove that the points of contact of tangents to the conics from a given point lie on an hyperbola passing through \(A\) and \(A'\).
Explain briefly the theory of any one form of dynamo, mentioning only the essential or most important features of it.
Shew, by means of the transformation \((1-\cos\theta\cos x)(1+\cos\theta\cos y) = \sin^2\theta\), or otherwise, that \[\int_0^\pi \frac{dx}{1-\cos\theta\cos x} = \frac{\pi}{2\sin\theta}, \quad \int_0^\pi \frac{dx}{(1-\cos\theta\cos x)^2} = \frac{\pi+2\cos\theta}{2\sin^3\theta}.\]