Problems

Assuming that the equation \[ x\frac{d^2y}{dx^2} + \frac{dy}{dx} - m^2xy = 0 \] is satisfied by a solution of the form \[ y = \sum_{r=0}^{\infty} c_r x^{\alpha+r}, \] where \(c_0\) is not zero, find the value of \(\alpha\), prove that \(c_{2r+1}=0\) and that \[ 4r^2c_{2r} = m^2c_{2r-2}, \] and write down the first four terms of the series.

A particle of mass \(m\) is placed on the centre of a plank of length \(2l\) and mass \(M\) which rests on a horizontal table. A horizontal impulse is then applied to the plank in the direction of its length. If the coefficients of friction between the plank and the table and between the plank and the particle are both \(\mu\), shew that the magnitude of the impulse which just suffices to knock the plank clear of the particle is \[ \{4\mu glM(M+m)\}^{\frac{1}{2}}. \]

A gun and an object fired at are in a horizontal plane, and the angle of elevation necessary to hit the object is \(\theta \left(<\frac{\pi}{4}\right)\). If the elevation is increased by \(\alpha\) the shot passes through a point at a vertical distance \(k\) from the object; if the vertical plane of projection is rotated through the same angle \(\alpha\) about the vertical through the gun the shot hits the plane at a distance \(c\) from the object. Find the value of \(\frac{k}{c}\) in terms of \(\theta\) and \(\alpha\), and shew that, if \(\alpha\) is small, \[ k=c(1-\tan^2\theta - 2\alpha \tan^3\theta), \text{ approximately}. \]

The lines joining a point \(P\) on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) to the points \((\pm c, 0)\) meet the ellipse again in \(Q\) and \(R\). The tangents at \(Q\) and \(R\) meet in \(T\). Prove that \(PT\) is bisected by the minor axis.

Prove that, if \[ x=r \sin\theta \cos\phi, \quad y=r \sin\theta \sin\phi \quad \text{and} \quad z=r \cos\theta, \] then \[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right)^n \frac{1}{r^m} = \frac{(m-1)m(m+1)(m+2)\dots(m+2n-2)}{r^{m+2n}}, \] and find the value of \[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) \log \tan \frac{1}{2}\theta. \]

The point of suspension of a simple pendulum initially at rest is made to move in a horizontal straight line with constant acceleration \(g/2\). If the bob (weight \(W\)) is initially at rest and vertically below the point of suspension, prove that throughout the motion the inclination of the string to the vertical is always less than \(54^\circ\), and find the tension in the string when the inclination is \(30^\circ\).

A bead of mass \(A\) can slide freely on a horizontal wire and is attached to a mass \(B\) by a light inextensible string of length \(l\). When \(A\) is at rest, the mass \(B\) is let fall from rest at a point on the wire distant \(l\) from \(A\). Determine the velocities of \(A\) and \(B\) when the line joining the masses is vertical.

\(AB\) is a fixed diameter of a rectangular hyperbola and \(P\) a variable point on the hyperbola. Prove that the difference between the angles \(PAB\) and \(PBA\) is equal to one of two fixed supplementary angles. Hence or otherwise prove that the two circles touching the hyperbola at \(P\) and passing respectively through \(A\) and \(B\) are of equal radius.

Prove that, if \(r, r'\) denote the distances of a point \(P\) from two fixed points \(A, B\) and \(\theta, \theta'\) the angles that \(AP, BP\) make with a fixed direction, then the families of curves \[ r^m r'^n = \alpha, \] \[ m\theta + n\theta' = \beta, \] where \(m\) and \(n\) are constants and \(\alpha, \beta\) are variable parameters, cut one another orthogonally.

A light rod 4 ft. long is free to rotate about one end which is fixed and carries a massive particle at the other end. The rod is kept in equilibrium in a horizontal position by a light inextensible string which joins the middle point of the rod to a fixed point 3 ft. vertically above the particle. Prove that small oscillations of the system are of period 1.92 sec. approximately.