Problems

Shew that, if \[ e^x \sin x = a_0 + \frac{a_1}{1!}x + \frac{a_2}{2!}x^2 + \dots + \frac{a_n}{n!}x^n + \dots, \] then \(a_{4n}=0\), \(a_{4n+1} = (-1)^n 4^n\); and determine \(a_{4n+2}\) and \(a_{4n+3}\).

Two coplanar forces \(X, Y\) are parallel to the (rectangular) axes of \(x\) and \(y\) respectively, their points of application being \((a,0)\) and \((0,b)\). If the forces are rotated through the same angle, shew that their resultant always passes through the fixed point whose coordinates are \[ x = \frac{aX^2+bXY}{X^2+Y^2}, \quad y=\frac{bY^2+aXY}{X^2+Y^2}. \]

A smooth sphere of mass \(m\) is resting on a smooth horizontal inelastic table. A second sphere of mass \(M\) is dropped vertically so as to strike the first with velocity \(V\), the line of centres at the moment of impact making an angle \(\theta\) with the vertical. If the coefficient of restitution between the spheres is \(e\), show that the velocity communicated to the first sphere is \[ (1+e)MV\frac{\sin\theta\cos\theta}{m+M\sin^2\theta}. \]

From the point \(P(\alpha\cos\theta, \beta\sin\theta)\) of the ellipse \(S' \equiv \frac{x^2}{\alpha^2} + \frac{y^2}{\beta^2} - 1 = 0\) tangents are drawn to the ellipse \(S \equiv \frac{x^2}{a^2} + \frac{y^2}{b^2} - 1 = 0\) to meet \(S'\) again in points \(Q, R\). \par Shew that the equation of the line \(QR\) is \[ \frac{x}{\alpha}(a^2\beta^2+a^2b^2-\alpha^2b^2)\cos\theta + \frac{y}{\beta}(\alpha^2\beta^2-a^2b^2+\alpha^2b^2)\sin\theta + (a^2b^2-\alpha^2b^2-a^2\beta^2)=0. \]

Find the maxima and minima values of \[ \frac{x+y-1}{x^2+2y^2+2}. \]

A light rod of length \(a\) has at one end a particle, and at the other end a smooth ring of equal mass, which is free to move along a straight horizontal wire. The rod is held with the particle in contact with the wire, and is then released. Shew that, when the rod has turned through an angle \(\theta\), the velocity of the particle is \[ \left\{ag\sin\theta\frac{4-3\sin^2\theta}{2-\sin^2\theta}\right\}^{\frac{1}{2}}. \]

A train of mass 300 tons has a driving force of 5 tons weight, and the resistances are \(v^2/1000\) tons weight, where \(v\) is the velocity in miles per hour. If the train is started from rest up an incline of 1 in 300, find the velocity at the end of a mile. \[ [\log_{10}e = 0.4343.] \]

A rectangular hyperbola \(H\) with centre \(O\) cuts a line \(l\) in two points \(P, Q\). \(X\) is the pole with respect to \(H\) of the line joining the remaining intersections of \(H\) with the circle \(OPQ\). \(O\) and \(l\) are fixed and \(H\) is allowed to vary. Shew that \(X\) remains fixed.

Trace the curve \[ y^4 - 4axy^2 + 3a^2x^2 - x^4 = 0, \] and shew that it has tangents parallel to the axis of \(x\) at two points for which \(x^4 = \frac{3}{4}a^4\).

A box of mass \(M\) rests on a rough horizontal table and from the centre of the lid of the box there hangs a pendulum of length \(l\) whose bob has mass \(m\). The pendulum vibrates through a right angle on either side of the vertical. Assuming that the box does not tilt, shew that it does not slide on the table if the coefficient of friction between box and table is greater than \[ \frac{3m}{2\sqrt{M(M+3m)}}. \] If the table had been smooth, shew that the box would have oscillated through a distance \[ \frac{2ml}{M+m}. \]