Prove that two couples, acting in one plane upon a rigid body, are in equilibrium if their moments are equal and in opposite senses. A uniform bar of weight \(W\) and length \(2a\) is suspended, from two points in a horizontal plane, by two equal strings of length \(l\), which are originally vertical: shew that the couple, which must be applied to the bar in a horizontal plane, to keep it at rest at right angles to its former direction is \[ \frac{Wa^2}{\sqrt{l^2-2a^2}}. \]
State the laws of friction and shew how they may be verified experimentally. A weight is pulled up a rough inclined plane by a rope parallel to a line of greatest slope; find the mechanical advantage and the efficiency of this machine.
Explain the terms Stable, Unstable and Neutral Equilibrium. A solid circular cylinder of radius \(a\) and height \(h\) has one end in the shape of a hemisphere; find the condition that it will be in stable equilibrium when standing on that end, on a smooth horizontal plane, with its axis vertical.
Reciprocate, with respect to the focus, the theorem that the circumcircle of the triangle formed by three tangents to a parabola passes through the focus. Generalise both theorems by projection.
Shew that the necessary and sufficient conditions that both roots of the equation \[ x^2+ax+b=0 \] should have their modulus equal to unity, are \[ |a|\leq 2, \quad |b|=1, \quad am b = 2 \text{ am } a. \] % "am" likely means "argument" or "amplitude"
Solution: (\(\Rightarrow\)) Suppose the roots are \(z_1, z_2\) then \(|b| = |z_1z_2| = |z_1||z_2| = 1\). \(|a| = |-(z_1+z_2)| \leq |z_1|+|z_2| = 2\) \(\arg(b) = \arg(z_1)+ \arg(z_2)\). \(\arg(a) = \arg(\cos \theta_1+\cos \theta_2 + i(\sin \theta_1 + \sin \theta_2)) = \arg (2 \cos\frac{\theta_1+\theta_2}{2}\cos \frac{\theta_1-\theta_2}{2} + 2i \sin \frac{\theta_1+\theta_2}{2} \cos \frac{\theta_1 - \theta_2}{2}) = \frac{\theta_1+\theta_2}{2}\)
Discuss the convergence of the series \[ \sum \frac{1 \cdot 3 \cdot 5 \dots (2n+1)}{2 \cdot 4 \cdot 6 \dots (2n+2)} p^n x^n \] for all real values of \(x\) and \(p\).
Prove that, if \(f(x)\) is continuous for \(a \leq x \leq b\), then \[ \frac{1}{n} \sum_{\nu=0}^{n-1} f\left\{a+\frac{\nu}{n}(b-a)\right\} \] tends to a limit as \(n\to\infty\). Taking \(a=0, b=1\), and \(f(x)=x\log x\), deduce that \[ 1^1 2^2 \dots (n-1)^{n-1} = n^{\frac{1}{2}n^2} e^{-\frac{1}{4}n^2(1+\epsilon_n)}, \] where \(\epsilon_n \to 0\) as \(n \to \infty\).
Shew that the integral \[ \int_0^\pi \cos(x\sin\phi-n\phi)d\phi \] is a solution of the equation \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+(x^2-n^2)y=0, \] if and only if \(n\) is an integer.
Shew that the circles \[ (x-a)^2+(y-b)^2=r^2, \] where \(a, b\) and \(r\) are functions of a parameter \(t\), will be the circles of curvature of their envelope if \[ r'^2 = a'^2+b'^2, \] dashes denoting differentiations with respect to \(t\); and that the envelope will be given by \[ x = a - \frac{a'r}{r'}, \quad y = b - \frac{b'r}{r'}. \]
Shew that the equation of the osculating plane at the point \((x,y,z)\) of the sphero-conic in which the cone \(a\xi^2+b\eta^2+c\zeta^2=0\) cuts the sphere \(\xi^2+\eta^2+\zeta^2=1\) is \[ \frac{a(\xi-x)x^3}{b-c} + \frac{b(\eta-y)y^3}{c-a} + \frac{c(\zeta-z)z^3}{a-b} = 0. \]