Problems

If \[ y = (x+1)^\alpha (x-1)^\beta, \] prove that \[ \frac{d^n y}{dx^n} = (x+1)^{\alpha-n} (x-1)^{\beta-n} Q_n(x), \] where \(Q_n(x)\) is a polynomial of degree \(n\) (or lower) in \(x\), and shew that \[ Q_{n+1}(x) = \{(\alpha+\beta-2n)x - (\alpha-\beta)\}Q_n(x) + (x^2-1)Q_n'(x). \] Prove also that \[ (1-t)^n Q_n\left(\frac{1+t}{1-t}\right) = 2^n n! \sum_{v=0}^n \binom{\alpha}{v} \binom{\beta}{n-v} t^v, \] where \(\dbinom{\alpha}{0}=1\), \(\dbinom{\alpha}{v} = \dfrac{\alpha(\alpha-1)\dots(\alpha-v+1)}{v!}\) (\(v=1,2,\dots\)).

A simple pendulum of length \(l\) is initially at rest. Its point of suspension is suddenly set moving with constant velocity \(u\) in a horizontal straight line. Find the conditions (i) that the bob may describe complete circles relatively to the point of suspension, (ii) that the string may oscillate about the vertical remaining taut throughout.

From a fixed orifice \(m\) pounds of water issue per second with velocity \(V\) feet per second. The jet impinges at once upon a smooth flat plate whose plane is inclined so that its normal makes an angle \(\theta\) with the direction of the jet. If the plate is moved in the same direction as the jet with a velocity \(v\) feet per second and the water does not rebound from the plate, show that the power delivered by the jet to the plate is a maximum when \(v=\dfrac{V}{3}\) and obtain an expression for this maximum power.

Obtain the equation of the polar of the point \(P(\xi, \eta)\) with respect to the conic \[ \frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1 \] and shew that as \(\lambda\) varies the polar of \(P\) envelopes a parabola touching the coordinate axes \(OX\) and \(OY\) and having \(OP\) as directrix. Shew also that the focus \(Q\) of this parabola is situated at the point \((\mu\xi, -\mu\eta)\), where \[ \mu = (a^2-b^2)/(\xi^2+\eta^2); \] deduce that as \(P\) describes any given circle, \(Q\) will also describe a circle.

The variables \((x,y)\) in \(f(x,y)\) are changed to \((\xi, \eta)\) by the substitution \[ x = \tfrac{1}{2}(\xi^2 - \eta^2), \quad y = \xi\eta, \] and \(f(x,y)\) becomes \(\phi(\xi, \eta)\). Prove that \[ \xi \frac{\partial\phi}{\partial\xi} + \eta \frac{\partial\phi}{\partial\eta} = 2 \left( x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} \right), \] \[ \frac{\partial^2\phi}{\partial\xi^2} + \frac{\partial^2\phi}{\partial\eta^2} = (\xi^2+\eta^2)\left(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\right). \]

A particle moving in a plane is acted on by a repulsive force from a fixed point \(O\) of the plane, the force per unit mass being of magnitude \(n^2 r\) (\(n\) constant) when the particle is at distance \(r\) from \(O\); prove that the path of the particle is a hyperbola. If initially the particle is at a point \(A\) distant \(a\) from \(O\) and is moving with velocity \(u\) in a direction which makes an angle \(\alpha \left(<\frac{\pi}{2}\right)\) with \(OA\) produced, prove that the particle will ultimately move in a direction which makes an angle \[ \tan^{-1}\left[\frac{na\sin\alpha}{na\cos\alpha+u}\right] \] with the direction of its initial velocity.

A particle of mass \(m\) is dropped from rest and impinges with velocity \((2gh)^{\frac{1}{2}}\) on a point \(A\) of the smooth face of a wedge of mass \(M\) and slope \(\alpha\) which is at rest and free to move on a smooth inelastic horizontal plane; the wedge is symmetrical about a vertical plane through the initial position of the particle. Shew that (the face of the wedge being supposed large enough) the particle will next strike a point \(B\) of the wedge distant \[ \frac{4e(1+e)(M+m)}{M+m\sin^2\alpha} h \sin\alpha \] from \(A\), where \(e\) denotes the coefficient of restitution between the wedge and the particle.

Obtain the equation of a normal to the hyperbola \[ x^2/a^2 - y^2/b^2 = 1 \] in the form \[ ax\sin\phi + by = (a^2+b^2)\tan\phi. \] Shew that if four normals to this hyperbola form the sides of a square, then the area of the square is \[ 2(a^2+b^2)^2/(a^2-b^2). \] Find the area of the square formed by the four tangents at the feet of these normals.

On the tangent at \(P\) to a plane curve \(\Gamma\) a point \(P'\) is taken so that \(PP'=a\), where \(a\) is a fixed positive number and \(PP'\) is drawn in the sense corresponding to increasing \(s\), where \(s\) is the arc of \(\Gamma\) measured from a fixed point to \(P\). As \(P\) describes \(\Gamma\) the point \(P'\) describes a curve \(\Gamma'\). If \(s'\) is the arc of \(\Gamma'\) measured from a fixed point to \(P'\), shew that \[ \frac{ds'}{ds} = \frac{R}{\rho}, \] where \(\rho = CP\) and \(R=CP'\), \(C\) being the centre of curvature of \(\Gamma\) at \(P\). Prove also that the centre of curvature of \(\Gamma'\) at \(P'\) is the point \(C'\) on \(CP'\) such that \[ \frac{C C'}{C'P'} = -\frac{a\rho}{R^2} \frac{d\rho}{ds}. \]

Two thin uniform rods \(AB\) and \(BC\), each of mass \(m\) and length \(l\), are smoothly hinged together at \(B\) and are supported in a vertical plane with their ends \(A\) and \(C\) resting on a fixed smooth horizontal plane. The angle \(ABC\) is 60\(^\circ\). If the system is released so that it is free to collapse in a vertical plane, determine the velocity with which the hinge \(B\) will strike the horizontal plane and the resultant impulse which brings the system to rest, assuming that there is no rebound.