Problems

If \(y = e^{ax^2}\) and \(u = \frac{d^n y}{dx^n}\), prove that \[ \frac{d^2u}{dx^2} - 2ax \frac{du}{dx} - (2n+2)au = 0. \] If \(u = e^{ax^2}v\), find a differential equation satisfied by \(v\). Shew that \(v\) is a polynomial of degree \(n\) in \(x\), and find the coefficient of \(x^{n-2}\).

A cylinder A rolls without slipping on the outside of a fixed horizontal cylinder B, the generators remaining parallel. A is slightly disturbed from equilibrium in a position in which the common tangent plane is horizontal. Shew that the equilibrium is stable if \[ 1/h > 1/r_1 \pm 1/r_2, \] where \(h\) is the height of the centre of gravity of the body above the point of support, and \(r_1, r_2\) are the radii of curvature of the body and the fixed surface. What is the significance of the choice of sign? A uniform elliptic cylinder of semi-axes \(a\) and \(b\) (\(a > b\)) is placed on the top of a fixed rough circular cylinder of radius \(r\), the generators being parallel and horizontal. Shew that there are positions of equilibrium when the major axis of the cross-section is either horizontal or vertical, but that the second of these is always unstable. Shew also that the position in which the major axis is horizontal is stable only if \[ r > a^2b/(a^2 - b^2). \]

A particle is free to move on a smooth vertical circle of radius \(a\). It is projected from the lowest point with velocity just sufficient to carry it to the highest point. Shew that, after a time \[ \sqrt{\frac{a}{g}} \log_e (\sqrt{5} + \sqrt{6}), \] the reaction between the particle and the wire is zero.

Prove that the mid-points of the sides of a triangle inscribed in a rectangular hyperbola \(H\) lie on a circle through the centre of \(H\).

Sketch the curve \[ y^2 = \frac{2x-1}{x^2-1}. \] Shew that \(x+y=1\) is an inflexional tangent. Are there any others?

Two uniform circular cylinders of the same radius rest on an inclined plane and touch along a generator, their axes being horizontal. All the surfaces are rough, with the same coefficient of friction \(\mu\). Shew that, for equilibrium to be possible, the conditions \[ W_2 > W_1, \quad W_2(2\cot\alpha-1) > W_1 \] must be satisfied, where \(W_1\) is the weight of the lower cylinder, \(W_2\) that of the upper cylinder, and \(\alpha\) is the inclination of the plane to the horizontal. If these conditions are satisfied, find the least value of \(\mu\) which makes equilibrium possible for a given \(\alpha\).

A train of mass \(M\) is moving with velocity \(V\) when it begins to pick up water at a uniform rate. The power is constant and equal to \(H\). If after time \(t\) a mass \(m\) of water has been picked up, find the velocity and shew that the loss in energy is \[ \frac{m(Ht + MV^2)}{2(m+M)}. \]

Shew that chords of a conic \(S\) which subtend a right angle at a given point \(O\) of \(S\) pass through a fixed point \(F\) lying on the normal at \(O\). If \(S\) is a parabola, shew that as \(O\) describes \(S\), the point \(F\) describes an equal parabola.

Prove that the evolute of the logarithmic spiral \(r=ae^{\alpha\theta}\) is an equal spiral.

A bullet of mass \(m\) is fired horizontally with velocity \(V\) into a block of mass \(M\) which rests on a rough horizontal plane, the coefficient of friction being \(\mu\). The block offers a constant resistance \(F\) to the bullet so long as there is any relative motion. Shew that, if \(F > \mu(M+m)g\), the bullet penetrates a distance \[ \frac{mM}{2(m+M)} \frac{V^2}{(F - \mu mg)} \] into the block, and find the time which elapses before the block comes to rest.