Obtain the equation of a tangent to the parabola \(l/r = 1+\cos\theta\) in the form \(l/r = \cos\theta + \cos(\theta-\alpha)\). Prove that the equation of the common tangent to the parabolas \[ 2a/r = 1+\cos(\theta-\gamma) \text{ and } 2b/r = 1+\cos(\theta+\gamma) \] is \[ \frac{a^2-2ab \cos 2\gamma + b^2}{r\sin 2\gamma} = a\sin(\theta+\gamma) + b\sin(\gamma-\theta). \]
Two particles of masses \(m\) and \(3m\) are connected by a fine string passing over a fixed smooth pulley. The system starts from rest and the heavier particle, after falling 8 feet, impinges on a fixed inelastic support. Find the velocity with which it is next jerked off the support; and shew that the system finally comes to rest 3 seconds from the beginning of the motion.
In a triangle \(ABC\) the side \(AB\) and the distance from \(C\) to the middle point of \(AB\) are accurately measured and are 60 and 30.04 inches respectively. \(AC, BC,\) are more roughly estimated and are found to be nearly equal; shew that the angle \(C\) is approximately 69" less than a right angle.
State the laws of Conservation of Linear Momentum and of Conservation of Energy. Shew that in an inelastic impact there is always loss of kinetic energy. A chain is formed of \(n\) equal particles attached at regular intervals to a light string. If the chain is heaped on a table and lifted off the table by a vertical force equal to its weight, shew that the energy that has been dissipated when the chain is clear of the table is \(\frac{1}{3}\left(1-\frac{1}{2n}\right)\) times the work done by the force.
Give a definition of the unit current in the C.G.S. system in terms of the unit magnetic pole, and shew by considering the latter unit how the former depends only on the fundamental units of length, time and mass. Given that the C.G.S. unit of current is 10 amperes, find the magnetic force at a point on the axis of a coil having 15 turns of 10 cm. radius and carrying a current of 25 amperes, the point being 20 cm. from the plane of the coil.
If the tangents to an ellipse from a point \(T\) touch at \(P\) and \(Q\), and if \(N\) is the foot of the perpendicular from \(T\) on the major axis, prove that \(TP\) and \(TQ\) subtend equal or supplementary angles at \(N\) according as \(N\) lies inside or outside the ellipse.
A particle is projected with given velocity from a point \(P\) so as to pass through a point \(Q\). If \(S\) is the focus of either of the possible trajectories, shew that the times of flight in the two trajectories are \(\{(SP+SQ+PQ)^{\frac{1}{2}} \pm (SP+SQ-PQ)^{\frac{1}{2}}\}/g^{\frac{1}{2}}\).
Shew that the error in taking \(\frac{3\sin\theta}{2+\cos\theta}\) for \(\theta\) is less than two-thirds per cent. when \(\theta\) is less than a radian.
Prove that any displacement of a rigid lamina in its own plane can be effected by a curve fixed in the lamina rolling on a curve fixed in the plane. Shew that if two given points of the lamina describe straight lines, any point on a certain circle fixed in the lamina will also describe a straight line.
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.