A wire framework consists of 10 equal wires, each of resistance 1 ohm, placed so that they form three squares side by side. Find the resistance of the framework between diagonally opposite corners.
The lines joining any point \(P\) on the ellipse \(x^2/a^2+y^2/b^2=1\) to the points \((\lambda a, 0)\) and \((\lambda' a, 0)\) meet the ellipse again in \(Q\) and \(R\). Shew that the envelope of the chord \(QR\) for all positions of \(P\) is the conic \[ \frac{x^2}{a^2} + \frac{1-\lambda\lambda'}{(1-\lambda)(1-\lambda')}\frac{y^2}{b^2}=1. \] Discuss the case in which \(\lambda\lambda'=1\).
Given a curve, drawn on a distance base, representing the velocity of a moving point, shew that the linear acceleration in any position is represented by the subnormal of the curve. If the curve is drawn to scales such that \(1''\) represents \(x\) ft., and \(1''\) represents \(y\) ft. per sec., find the scale on which the acceleration is to be interpreted.
Two orthogonal circles meet in \(A, A'\) and their common tangents meet at \(T\). If \(AT\) makes angles \(\alpha, \beta\) with the common tangents and if \(2\theta\) is the angle between these common tangents, shew that \[ 2 \sin\alpha \sin\beta = \sin^2\theta. \]
Write an essay on simple harmonic motion. State and prove the necessary and sufficient relation between the displacement of a system from the equilibrium position, and the corresponding restoring force, in order that the system may vibrate with simple harmonic motion. Shew that the period is a property of the system, and is independent of the initial circumstances of displacement. Indicate the variation of velocity, acceleration and energy (kinetic and potential) throughout the course of an oscillation. Illustrate this by the case of a simple seconds pendulum of mass \(m\), which is drawn aside through a small angle \(\theta\) and then projected towards its mean position with a small angular velocity \(\omega\).
A steady P.D. of 5 volts is applied to a coil of copper wire which has a resistance of 100 ohms at 0\(^\circ\)C., and is such that it can radiate \(\frac{1}{100}\) watt per degree Centigrade rise of temperature above the atmospheric temperature (15\(^\circ\)C.). Find the final steady temperature, if the temperature coefficient of copper be 0.004 per degree Centigrade.
Find the asymptotes of the curve \[ x^2(x+y)=x+4y, \] and trace the curve.
A particle of mass \(m\) slides down the smooth inclined face (inclination \(\alpha\)) of a wedge of mass \(M\), placed on a rough horizontal table. Shew that, if the wedge slips on the table, the coefficient of friction, \(\mu\), between it and the table must be less than \(m \sin\alpha \cos\alpha / (M+m\cos^2\alpha)\); and that the pressure on the table is then \[ \frac{(M+m)Mg}{M+m(\sin^2\alpha - \mu \sin\alpha\cos\alpha)}. \]
From the vertex of the parabola \(y^2-4ax=0\), lines are drawn parallel to the tangents to the curve, prove that the locus of the points where they meet the corresponding normals is the curve \(y^4(x^2+y^2) = ax(x^2+2y^2)\).
State and prove the acceleration property of the hodograph. Determine the hodographs of (1) a projectile describing a parabolic path, (2) a particle on the inside of the rim of a cartwheel which is travelling with uniform speed along a straight road. The thickness of the rim is to be taken as \(\frac{1}{10}\) of its external radius. In each case draw also the actual paths of the bodies, and indicate corresponding points on path and hodograph. A radial force of \(1\frac{1}{2}\) ounces weight is needed to detach the particle, weighing 2 oz., from the rim. If the cart is travelling at 4 miles an hour, and the particle is in limiting equilibrium at the top of its path, find the radius of the wheel.