Problems

If \[ f_n(x) = e^x \frac{d^n}{dx^n}(x^n e^{-x}), \] prove that \[ x\frac{d^2f_n(x)}{dx^2} + (1-x)\frac{df_n(x)}{dx} + n f_n(x) = 0. \]

Evaluate \[ \int_0^\infty \frac{dx}{(x+1)\sqrt{(5x^2+12x+8)}}. \]

A circle of radius \(r\) is rotated through 180\(^\circ\) about an axis which lies in the plane of the circle and is at a perpendicular distance \(a(>r)\) from the centre of the circle. Find the mass-centre of a uniform solid occupying the volume generated by the circle, and explain why it does not coincide with the mass-centre of a uniform semi-circular wire occupying the path traced out by the centre of the circle.

A smooth rectangular plank of mass \(M\) fits accurately in a smooth horizontal groove along which it is free to slide. The plank has a deep rectangular groove with vertical sides cut in its upper face, the edges of the groove being parallel to the edges of the plank, and in this groove a ball of mass \(m\) fits closely and runs freely. With the plank at rest the ball is projected with velocity \(V\) along the groove. The coefficient of restitution between the ball and the ends of the groove is \(e\). Find an expression for the velocities of the plank and the ball just after the \(n\)th impact. Shew that the velocities of the plank and the ball both tend to the same limit.

The horizontal and inclined faces of a wedge of mass \(M\) meet at an angle \(\alpha\) in a line \(AB\). The smooth horizontal face has a smooth groove cut in it perpendicular to \(AB\), and the wedge rests on a smooth table, the groove running on a smooth rail fixed to the table so that any motion of the wedge is perpendicular to \(AB\). A particle of mass \(m\) slides down the rough inclined face of the wedge. Shew that, if the system starts from rest, the time taken by the particle to describe a given distance down the wedge bears to the time taken when the wedge is fixed the ratio \[ \left\{1 - \frac{m \cos\alpha(\cos\alpha+\mu\sin\alpha)}{M+m}\right\}^{\frac{1}{2}}:1, \] where \(\mu\) is the coefficient of friction between particle and wedge.

A particle placed close to the vertex of a smooth cycloid whose axis is vertical and vertex upward is allowed to run down the curve under gravity. Shew that it leaves the curve when it is moving in a direction making an angle of 45\(^\circ\) with the horizontal.

A uniform rod \(AB\) of mass \(M\) and length \(l\) hangs vertically down from a smooth hinge \(A\). When the rod is at rest an impulse is applied in a horizontal direction to a point \(C\) of the rod. Determine the position of \(C\) in order that there may be no impulsive reaction at \(A\), and then determine the magnitude of the impulse required in order that the rod may just reach the upward vertical position.

A, B, C are three points in a straight line. Three semicircles are constructed on AB, BC and AC as diameters, all on the same side of the line ABC, and a circle is drawn touching the three semicircles. Prove that its diameter is equal to the perpendicular distance of its centre from ABC.

\(F_1, F_2, F_3 \dots F_n\) are fixed coplanar forces. A new force \(F_{n+1}\) is added, whose point of application \(A\) and line of action are fixed, but whose magnitude can be varied. Shew that the resultant of the forces \(F_1, F_2, F_3 \dots F_{n+1}\) always passes through another fixed point \(B\), and by a suitable choice of \(F_{n+1}\) may be made to pass through any arbitrary point \(C\), which does not lie on the line \(AB\). Discuss any exceptional cases. \(PQRS\) is a square of side \(a\), and forces 1, 2, 3 act along \(PQ, QR, RS\) respectively. A variable force \(F\) acts along \(SP\). Find the fixed point through which the resultant always acts, and the value of \(F\) if it is to pass through the centre of the square.

\(OABC, OA'B'C'\) are two straight lines; \(AB', BA'\) meet at \(P\); \(BC', CB'\) meet at \(Q\), and \(CA', AC'\) meet at \(R\). Shew that \(P, Q, R\) lie on a straight line. Prove that, if the cross-ratios \(\{OB, AC\}\) and \(\{OB', A'C'\}\) are equal, the line \(PQR\) passes through \(O\).