Shew that if \(\sin n\theta\) is given, \(2n\) values of \(\sin\theta\) are to be expected if \(n\) is even and \(n\) if \(n\) is odd. If \(\frac{\sin(\alpha+\theta)}{\sin(\alpha+\phi)} = \frac{\sin(\beta+\theta)}{\sin(\beta+\phi)}\) shew that either \(\alpha\) and \(\beta\) or \(\theta\) and \(\phi\) differ by a multiple of \(\pi\).
Two spheres of masses \(m_1\) and \(m_2\) are in motion without rotating. Shew that the total kinetic energy is that of a mass \((m_1+m_2)\) moving with the velocity of their centre of mass, together with that of a mass \(m_1m_2/(m_1+m_2)\) moving with the velocity of separation of their centres. Hence determine the loss of kinetic energy in the oblique impact of two smooth elastic spheres. Prove that, if two smooth spheres are in motion with equal and opposite momenta, the effect of an impact is the same as if each sphere impinged on a fixed plane perpendicular to their lines of centres at the moment of impact.
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 \[ \frac{(M+m)Mg}{M+m(\sin^2\alpha - \mu \sin\alpha\cos\alpha)}. \]