A coil of copper wire, whose resistance is 50 ohms at 0° C., is immersed in water in a closed vessel: it is observed that when the temperature of the whole is 20° C. the rate of fall of temperature by radiation and conduction is 0.3° C. per minute. A constant P.D. is now applied to the coil and it is observed that when the temperature rises to 20° C. it is rising at the rate of 4.2° C. per minute: find the final steady temperature reached. The temperature coefficient of increase of resistance for copper is \(\cdot 004\) per degree C.; the atmospheric temperature is 15° C. throughout.
A uniform rod \(AB\) of length \(a\) can rotate about \(A\) in a vertical plane. It is supported in a horizontal position with the end \(B\) attached to an elastic string, the other end of which is fastened to a point vertically above \(B\). The string is just taut and of length \(l\). Show that, if the support is removed, the rod will turn through an angle \(\phi\) before coming to rest, where \[ a \sin \phi = b - a^2 (1 - \cos \phi)^2/l, \] approximately, \(a/l\) being small, and the elasticity of the string being such that the weight of the rod would produce an extension \(b\).
A curve is given by the equations \[ x = t^3, \quad y = t(t^2 - 5), \] \(t\) being a variable parameter. Give a sketch of the curve, showing its general form. Show in particular that there is a double point and find the tangents there. Prove that the curvature never changes sign.
State the laws of impact between smooth elastic spherical bodies; discuss the action between them, and shew how to find the changes in the velocities, and the impulses of compression and restitution. Find the loss of kinetic energy in oblique impact. Two unequal spherical masses are suspended from the ends of an inelastic string which passes over a smooth pulley. The two portions of the string are vertical and pass through small holes in a horizontal table, so that the masses hang below the table. Determine the change in the motion when one of the masses strikes the table with velocity \(v\), the coefficient of elasticity between the mass and the table being \(e\).
Find the equations of the axis and of the tangent at the vertex of the parabola given by the equation \[ 4x^2 + 12xy + 9y^2 + 10x + 2y - 10 = 0. \] Show that the focus is at \((1, -1)\), and find the equation of the directrix.
For a carbon filament electric lamp, a portion of the curve connecting P.D. and current is found to be a nearly straight line between 120 volts, \(\cdot 19\) amp. and 200 volts, \(\cdot 36\) amp. Two such lamps are put in parallel and are fed through a resistance of 50 ohms from a battery giving a constant P.D. of 180 volts. Find the total current taken.
A particle rests on a smooth horizontal table and is constrained by two springs, attached to fixed points in the plane of the table, whose tensions are \(\mu_1\) and \(\mu_2\) times their lengths. It is started in motion in any manner so as to remain on the table and left to itself. Show that the projection of the motion on any direction is simple harmonic with the same period. What is the most general form of the path of the particle?
Show that \(y = \frac{(x - \alpha)(x - \beta)}{x - \gamma}\) can take all values as \(x\) varies provided \(\gamma\) lies between \(\alpha\) and \(\beta\), and otherwise assumes all values except those included in an interval of length \(d\) where \[ d^2 = 16 (\alpha - \gamma) (\beta - \gamma). \]
A particle moves under gravity on a given smooth curve in a vertical plane; shew how to determine the velocity and the pressure on the particle at any point. The particle describes a complete circle and the pressure is always directed inwards; shew that, if the maximum pressure is to the minimum pressure as \(m\) to \(n\), the maximum velocity is to the minimum velocity as \((5m+n)^{\frac{1}{2}}\) to \((m+5n)^{\frac{1}{2}}\).
Prove that, if \(x\) lies between 0° and 180°, \(\cos x - \frac{1}{4} \cos 2x\) lies between \(-\frac{3}{4}\) and \(\frac{1}{2}\), and find from the tables the values of \(x\) for which it is \(\frac{1}{4}\).