Problems

Two uniform rods \(OA\), \(AB\), smoothly jointed at \(A\), hang under gravity from a fixed smooth hinge at \(O\); each rod is of mass \(m\) and length \(a\), and \(B\) is constrained to move on a smooth vertical wire passing through \(O\). The equilibrium of the system becomes unstable when the ends of a light elastic string of modulus of elasticity \(mg\) are attached to \(O\) and \(B\); find the largest possible natural length of the string.

From a point \(T\) on the directrix of a parabola tangents \(TP\), \(TQ\) are drawn, and the chord \(PQ\) meets the directrix at \(K\) and the diameter through \(T\) in \(R\). Prove that \[ SP.SQ = SR.SK, \] \[ PQ^2 = 4RS.RK, \] \(S\) being the focus of the parabola.

The sequence \(u_0, u_1, u_2, \dots\) is defined by \(u_0=0\), \(u_1=1\), \(u_n=u_{n-1}+u_{n-2}\) (\(n=2, 3, \dots\)). Obtain a general formula for \(u_n\), and shew that \(u_n\) is the integer nearest to \[ \frac{1}{\sqrt{5}}\left(\frac{\sqrt{5}+1}{2}\right)^n. \]

\(A\) and \(B\) are two points at opposite ends of a diameter of a rectangular hyperbola, and \(P\) is a point which moves on the hyperbola. Prove that \(\angle PBA - \angle PAB\) is constant as long as \(P\) remains on the same branch of the curve and does not pass through \(A\) or \(B\). Find the relations between the four possible values of \(\angle PBA - \angle PAB\). If \(A\) and \(B\) are two given points, prove that the locus of \(P\), such that \[ \angle PBA - \angle PAB = \alpha, \] where \(\alpha\) is a given constant angle and \(P\) remains on the same side of \(AB\), is part of a rectangular hyperbola, and find the position of the transverse axis in relation to \(AB\).

A uniform chain suspended from two points on the same level, hangs partly in air of negligible density and partly in liquid of which the density is half the (constant) density of the material of the chain. One-third of the length of the chain is immersed in the liquid, and the distance between the points where the chain leaves the surface of the liquid is half the distance between the supports; find the ratio of the depth of the liquid surface below the supports to the sag of the chain.

Prove that the tangent to an ellipse makes equal angles with the focal distances of the point of contact. \(P\) and \(P'\) are two points (not at opposite ends of a diameter) of an ellipse whose foci are \(S\) and \(S'\), such that \(SP\), \(S'P'\) are parallel. The tangent at \(P\) meets \(S'P'\) in \(Q'\), and the tangent at \(P'\) meets \(SP\) in \(Q\). Prove that \(QQ'\) is parallel to \(PP'\).

Express \[ \begin{vmatrix} 1 & a & a^3 \\ 1 & b & b^3 \\ 1 & c & c^3 \end{vmatrix} \] as a product of linear factors in \(a, b, c\). Hence, or otherwise, prove that, if \(\alpha + \beta + \gamma = \pi\), then \[ \begin{vmatrix} 1 & \sin\alpha & \sin 3\alpha \\ 1 & \sin\beta & \sin 3\beta \\ 1 & \sin\gamma & \sin 3\gamma \end{vmatrix} = -16 \sin\alpha \sin\beta \sin\gamma \sin\frac{\beta-\gamma}{2}\sin\frac{\gamma-\alpha}{2}\sin\frac{\alpha-\beta}{2}. \]

A curve \(C\) on the earth's surface (assumed to be a sphere of radius \(a\)) cuts the meridians at a constant acute angle \(\alpha\). Prove that \[ \cos\lambda \cosh m\phi = 1, \] where \(m=\cot\alpha\), and \((\lambda, \phi)\) are the latitude and longitude, respectively, of a variable point \(P\) of \(C\), \(\phi\) being measured continuously along \(C\) from the point where \(C\) crosses the equator. Find (i) the length of the arc of \(C\) between the points \(P_1(\lambda_1, \phi_1)\) and \(P_2(\lambda_2, \phi_2)\), and (ii) the area swept out by the projection of \(OP\) on the equatorial plane, when \(P\) describes the arc \(P_1P_2\) of \(C\). Give each result in terms of the latitudes \(\lambda_1\) and \(\lambda_2\).

A particle of mass \(m\) is suspended from a fixed point \(A\) by a light inextensible string of length \(a\) and is hanging at rest at \(B\) when it receives a horizontal impulse of magnitude \(m\left(\frac{4}{3}ag\right)^{\frac{1}{2}}\). Shew that in the subsequent motion the string becomes slack, the particle reaches a height \(27a/16\) above \(B\), and that when the string next becomes taut the particle is at \(B\).

\(PSP'\), \(QSQ'\) are any two focal chords of a parabola. Shew that the common chord of the circles described on \(PP'\), \(QQ'\) as diameters passes through the vertex of the parabola.