The area of a triangle is to be calculated from measurements of the side \(a\) and of the angles \(B\) and \(C\), but each of these angles is over-estimated by \(1'\). If the angle \(A\) is very nearly \(90^\circ\), shew that the resulting error in the area of the triangle is approximately \(0.00015 a^2\).
A curve \(C'\) is obtained by inverting the spiral \(r=ae^{m\theta}\) with respect to the circle with centre \((-a,0)\) and radius \(a\sqrt{2}\). Shew that, if \(s'\) is the arc of \(C'\) measured from the origin to the point \(P'\) which corresponds to the point `\(\theta\)' of the spiral, then \[ \frac{ds'}{d\theta} = \frac{a\sqrt{1+m^2}}{\cosh m\theta + \cos\theta}. \] Prove that \(C'\) is symmetrical about the origin, and draw a rough sketch of the curve.
An electric motor which gives a uniform driving torque drives a pump for which the torque required varies with the angle during each revolution according to the law \(T \propto \sin^2\theta\): the mean speed of the pump is 600 rev. per min. and the mean horse-power required is 8. To limit the fluctuation of speed during each revolution, a flywheel is provided between the motor and the pump which successively stores and gives out energy. Shew that the energy thus successively stored and given out by the flywheel is approximately 70 foot pounds.
Evaluate \[ \int \frac{x-1}{(x+1)(x^2+x+1)}\,dx, \quad \int \frac{1}{\sqrt{x}}\sqrt{\frac{1-x}{1+x}}\,dx, \quad \int \frac{\log x}{x}\,dx. \]
Shew that if \(n\) is a positive integer, then \[ \frac{\sin(2n+1)\theta}{\sin\theta} = c_0 + 2c_1\cos 2\theta + 2c_2\cos 4\theta + \dots + 2c_n\cos 2n\theta, \] where \(c_r\) is the coefficient of \(x^r\) in \[ (x^{-n} + \dots + x^{-1} + 1 + x + \dots + x^n)^2. \] Hence, or otherwise, prove the formula \[ \int_0^{\frac{1}{2}\pi} \frac{\sin(2n+1)\theta}{\sin\theta} d\theta = \frac{1}{2}(2n+1)(8n^2+8n+3)\pi. \]
A hollow sphere, of internal radius 5 inches, spins with uniform angular velocity about a vertical axis through its centre. It carries round with it a rough heavy particle, which rests in contact with a point on its inner surface 3 inches below its centre and 4 inches from the axis of rotation. The speed of rotation is such that the particle is on the point of slipping down, whilst the centrifugal force upon it is equal to its weight. In what proportion must the speed be raised in order that the particle may be on the point of slipping up?
A plane area is formed of the circle \(r=a\) and the portions of the four loops of the curve \(r=2a\sin 2\theta\) exterior to the circle. Shew that the whole area of the figure is \((\frac{2}{3}\pi + \sqrt{3})a^2\).
Prove that, if a body is in equilibrium under three forces, the lines of action of the three forces are coplanar and either meet in a point or are parallel. The mass per unit length of a ladder increases uniformly from the top to the bottom of the ladder, and is twice as great at the bottom as at the top. The ladder stands on a horizontal plane and rests against a vertical wall. The angle of friction at both ends of the ladder is \(\epsilon\). If the ladder is just about to slip, prove that its inclination \(\theta\) to the horizon is given by \[ \tan\theta = \frac{5}{6}\cot\epsilon - \frac{2}{3}\tan\epsilon. \]
State the principle of virtual work. A weightless tripod, consisting of three legs of equal length \(l\), smoothly jointed at the apex, stands on a smooth horizontal plane. A weight \(W\) hangs from the apex. The tripod is prevented from collapsing by three inextensible strings, each of length \(l/2\), joining the mid-points of the legs. Shew that the tension in each string is \(\displaystyle\frac{\sqrt{2}}{3\sqrt{3}}W\).
Define mechanical advantage and efficiency. Shew that the mechanical advantage in the pulley system shown is twice the efficiency, the radii of the pulleys being \(a\) and \(a/2\), and \(A\) being a fixed point directly below the axis of the upper pulley. The lower pulley runs smoothly on its bearings, whereas the rotation of the upper pulley is opposed by a frictional couple proportional to the pressure of the pulley on its bearings. If \(E\) is the efficiency, shew that the least force \(P\) which will just prevent the weight from slipping down is \(\frac{1}{2}EW\). (Neglect the weight of the upper pulley and assume that the rope does not slip.)