Problems

A perfectly rough uniform plank of thickness \(t\) rests horizontally on the top of a fixed circular cylinder of radius \(a\) whose axis is horizontal and perpendicular to the long edges of the plank. Show that equilibrium is stable if \(t<2a\). Investigate the case \(t=2a\). If \(t<2a\), find an equation for the greatest angular displacement which can be made consistent with the plank returning towards the horizontal position if released.

A uniform horizontal beam which is to carry a uniformly distributed load is supported at one end and at one other point. Assuming that the beam will break if the bending moment at any point exceeds a certain value, show that, in order that the greatest possible load may be carried by the beam, the second support must be placed at a point which divides the beam in the ratio \(1:(\sqrt{2}-1)\).

An engine and train of total mass \(M\) move on horizontal rails, the pull of the engine being constant. The total resistance to the motion is equal to \(fv/v_0\) per unit mass, where \(v\) is the speed of the train, \(v_0\) is its maximum speed, and \(f\) is a constant. The train is moving with its maximum speed when the last coach, whose mass is \(m\), is slipped. Show that after a time \(v_0/f\) the rest of the train has moved a distance \[ \frac{v_0^2}{f}\left\{\ln\left(\frac{M}{M-m}\right) - \frac{m}{M}\right\}, \] where \(e\) is the base of Napierian logarithms.

A long light inextensible string passes over a light frictionless pulley and carries a bucket of mass \(M\) at one end and a counterpoise of mass \(M\) at the other. The bucket and counterpoise are in equilibrium when an elastic particle of mass \(m\) is dropped into the bucket so that it hits the horizontal bottom of the bucket with a velocity \(u\). Show that the particle ceases to bounce after a time \[ T = \frac{2eu}{g(1-e)}, \] where \(e\) is the coefficient of restitution between the particle and the bucket. By considering the fact that the total momentum of the system in the direction of the string is unaltered by impulsive tensions in the string, or otherwise, prove that the velocity of the bucket after time \(T\) is \[ \frac{1+e}{1-e}\frac{mu}{2M+m}. \]

A particle is attached by a light inextensible string of length \(a\) to a fixed point. The particle hangs in equilibrium and is then given a horizontal velocity \(\sqrt{(7ga/2)}\). Show that during the subsequent motion the maximum height of the particle above its initial position is \(27a/16\).

A particle of mass \(m\) hangs from a fixed point by an elastic string of natural length \(l\) and modulus of elasticity \(2mg\), and a second particle of the same mass hangs from the first by a similar string. The whole system lies in a vertical plane which rotates with uniform angular velocity \(\omega\) about the vertical through the point of suspension, the strings making constant angles \(\alpha\) and \(\beta\) with the vertical. Show that \begin{align*} \tan\alpha &= \frac{l\omega^2}{4g}\{4\tan\alpha+4\sin\alpha+\tan\beta+2\sin\beta\}, \\ \tan\beta &= \frac{l\omega^2}{2g}\{2\tan\alpha+2\sin\alpha+\tan\beta+2\sin\beta\}. \end{align*}

Three particles \(A, B, C\), each of mass \(m\), are connected by light inextensible strings \(AB, BC\), each of length \(a\). The particles are placed on a smooth horizontal table so that the strings are tight and the angle \(ABC\) is equal to \(90^\circ\). The particle \(B\) receives a blow of impulse \(I\) in the direction \(DB\), where \(D\) is the mid-point of \(AC\). Find the initial velocities of all the particles.

Give a short account of the method of generalisation by projection. Obtain the projective generalisations of concentric circles, right angle, rectangular hyperbola. Prove the following theorem. A fixed line through \(A\) meets at \(P\) a variable circle through two fixed points \(A, B\); the tangent at \(P\) to the circle envelops a parabola with focus \(B\). Generalise this theorem by projection. State the dual of the generalised theorem and obtain from it a result for a system of parabolas.

If a rigid body is in equilibrium under the action of two coplanar couples, deduce from the triangle (or parallelogram) of forces that the sum of the moments of the couples is zero. Show that forces \(1, 2, 3, 4, -11, 6\) acting along the sides \(AB, BC, CD, DE, EF, FA\), respectively, of a regular hexagonal lamina have a resultant acting along the line joining the mid-points of the sides \(AB, CD\), and find its magnitude.

Three collinear points \(A, B, C\) are given. Give a construction, making use of a ruler only, for the harmonic conjugate of \(C\) with respect to \(A\) and \(B\). Show how to construct, by using a ruler only, a line through a given point \(O\) to cut the sides \(YZ, ZX, XY\), of a given triangle in points \(P, Q, R\) respectively so that the range \((OPQR)\) may be harmonic.