A circle is inscribed in a \(60^\circ\) sector of a circle of radius \(a\). Find the radius of the inscribed circle and the areas of the various portions into which the sector is divided.
The director-circle and the directions of the axes of a variable conic are given. Find an equation for the envelope of the polar with respect to the conic of a fixed point on the director-circle, and show that it is a parabola touching the axes where the tangent to the circle at the fixed point meets them.
A weight is hung by two elastic strings from two points in the same horizontal line, the distance between the points being \(2b\). Each string is of unstretched length \(l\), and each would separately be stretched a distance \(a\) by the weight hanging vertically. If in the position of equilibrium the angle between the strings is \(2\theta\), shew that the periodic time of a small vertical oscillation about that position of equilibrium is that of a simple pendulum of length \[ \frac{a}{\frac{2}{l}\left\{1 - \frac{l}{b}\sin^3\theta\right\}}. \]
Discuss the friction between (1) a wheel of a vehicle in limiting equilibrium and its axle, assuming a loose bearing fit, (2) the wheel and the ground when (a) its weight is negligible compared with the load it carries, (b) its weight constitutes the greater part of the load, (c) a brake is directly applied to the wheel. Consider the resistance to the motion of a body which is placed on cylindrical rollers, and discuss your conclusions in connection with the assumptions usually made as to the nature of the contact between two bodies.
Shew that the tangents to the parabola \(y^2 = 4ax\) at the points where it is cut by the line \(rx + sy + a = 0\) include an angle \(\tan^{-1} \frac{2\sqrt{s^2-r}}{1+r}\).
A variable conic passes through the vertices of a triangle \(ABC\) and touches a given line through \(A\). Show that the locus of the centre is an ellipse or a hyperbola according as the given tangent divides \(BC\) internally or externally. Discuss in what circumstances (if any) the locus is a parabola or two straight lines.
A body is moving along a straight line; prove that the acceleration in any position is given by the gradient of the curve connecting \(\frac{1}{2}v^2\) and \(s\), where \(v\) is the velocity and \(s\) is the distance travelled. The speed of a motor cycle is observed, as it passes five posts placed 50 yards apart on a level track, to be 14.0, 26.4, 33.3, 37.1, 38.9 miles an hour respectively. Assuming the resistance in pounds weight due to mechanical and air friction to be \(6+0.02v^2\), where \(v\) is expressed in miles an hour, calculate the horse-power actually developed by the engine when the speed is 35 miles an hour, the total weight of machine and rider being 400 lbs. [\(g=32\), one H.P. = 33,000 ft. lbs. per min.].
Define Work, Power, Kinetic Energy, Potential Energy, Momentum. Prove any general theorems you know concerning them as applied to a system of particles, laying stress on the difference between conservative and non-conservative systems.
The tangent at the point \((4 \cos \phi, (16/\sqrt{11}) \sin \phi)\) to the ellipse \(16x^2 + 11y^2 = 256\) is also a tangent to the circle \(x^2 + y^2 - 2x = 15\). Find the values of \(\phi\).
Show that the curve \[ x^2(x+y) - y^2 = 0 \] has a cusp at the origin and the rectilinear asymptote \(x+y=1\), that no part of the curve lies between \(x=0\) and \(x=-4\), and that it consists of two infinite branches, one in the second quadrant and the other in the first and fourth quadrants. Give a sketch of the curve.