A uniform perfectly rough heavy plank of thickness \(2b\) rests symmetrically across the top of a fixed horizontal circular cylinder of radius \(a\), the length of the plank being perpendicular to the axis of the cylinder. Show that if the plank is turned without slipping through an angle \(\theta\) round the direction of the axis of the cylinder, the gain of potential energy is proportional to \[ a\theta\sin\theta - (a+b)(1-\cos\theta), \] and find the condition that the horizontal position shall be stable. Show further that if \(a>b\) there always exists a second position of equilibrium into which the plank can be turned.
A gun of mass \(M\) is free to recoil on a horizontal plane, and a shell of mass \(m\) is fired from it with the barrel elevated at an angle \(\alpha\). Show that if the muzzle velocity of the shell in space is \(v\), the horizontal range will be \[ 2v^2\beta/g(1+\beta^2), \] where \(\beta=(1+m/M)\tan\alpha\) and \(g\) is the acceleration of gravity.
A smooth sphere rests on a horizontal plane and is in contact with an inelastic vertical plane. An equal sphere moving on the horizontal plane with velocity \(v\) in a direction perpendicular to the wall strikes the first sphere and at impact the line of centres makes an angle \(\theta\) with the direction of \(v\). Find the speed communicated to the first sphere in terms of \(\theta\), and show that its greatest possible value is \(v(1+e)/2\sqrt{2}\), where \(e\) is the coefficient of restitution between the spheres.
A heavy particle of mass \(M\) rests on a smooth horizontal table at the centre of an equilateral triangle of side \(2a\), and three other particles each of mass \(m\) are attached to it by three strings that pass through holes in the table at the vertices of the triangle. The three particles hang vertically and initially the system rests in equilibrium. If one of the strings is suddenly cut, find the instantaneous change of tension in the other two and show that \(M\) begins to move with acceleration \(2mg/(2M+m)\). Find also the velocity of \(M\) when it crosses the side of the triangle.
A simple pendulum is making complete revolutions in a vertical plane in such a way that its greatest and least angular velocities are \(\omega_1\) and \(\omega_2\). Show that when the inclination of the pendulum to the downward vertical is \(\theta\) the angular velocity is \[ (\omega_1^2 \cos^2 \theta/2 + \omega_2^2 \sin^2 \theta/2)^\frac{1}{2}. \] Find the corresponding formula for the tension in terms of its greatest and least values, and hence show that critical values of the tension can never occur except at the highest and lowest points.
A straight rod \(OQ\) of length \(a\) rotates round \(O\) with constant angular velocity \(\omega\) so that \(Q\) describes a circle, and a rod \(QP\) of length \(b\) is freely jointed at \(Q\), while \(P\) is constrained to move in a straight line through \(O\). Show that when \(OQP\) are collinear, with \(Q\) between \(O\) and \(P\), the acceleration of \(P\) is of magnitude \(a(a+b)\omega^2/b\). Find also the acceleration of \(P\) when \(OQ\) is at right angles to \(OP\).
An engine and tender contain a quantity of fuel which is steadily consumed at the uniform rate of \(\mu\) units of mass per unit time. The engine thereby always does \(Q\) units of work per unit time and the resistance to its motion is \(k\) times the velocity. Assuming that the engine is moving on a level track, and denoting by \(M_0, v_0\) respectively the total mass and velocity at time \(t=0\), show that the velocity \(v\) at time \(t\) is given by the relation \[ Q - kv^3 = (Q-kv_0^3)(1-\mu t/M_0)^{2k/\mu}. \]
If two triangles are in perspective from a point, prove that the three points of intersection of pairs of corresponding sides lie in a straight line (the axis of perspective). Show that, if three triangles are in perspective, the axes of perspective of the three pairs of triangles are concurrent.
The coordinates of a point on a curve are \((at+bt^2, ct+dt^2)\), where \(t\) is a parameter. Prove that the equation of the chord joining the points \(t_1\) and \(t_2\) is \[ \begin{vmatrix} x & y & t_1t_2 \\ a & c & t_1+t_2 \\ b & d & -1 \end{vmatrix} = 0. \] If the tangents at the points \(t_1\) and \(t_2\) are at right angles, show that the chord passes through a fixed point, and find its coordinates.
A, B, C and D are the points of intersection of two conics S and S'. A variable line through A meets S in X, S' in Y and BC in Z. Prove that the locus of P, the harmonic conjugate of Z with respect to X and Y, is a conic passing through A, B, C and D.