Show that, in general, the resultant of a number of parallel forces of fixed magnitude acting at fixed points of a rigid body passes through a fixed point of the body whatever the common direction of the forces. State the conditions of the exceptional case, and show that in this case by adjusting the point of action of any one of the forces the system can be made one that is in equilibrium for all directions of the forces. Explain the relevance of the general case to the usual identification of centre of mass and centre of gravity.
A uniform heavy rod of length \(2b\) has its ends attached to small light rings which slide on a smooth rigid wire in the shape of a parabola of latus rectum \(4a\) held fixed in a vertical plane with its vertex uppermost. Prove that the horizontal position of the rod is one of stable equilibrium if \(b > 2a\). Show further that in this case there are two oblique positions of equilibrium, one on either side.
Explain what is meant by \emph{moment of inertia}. Show that for a plane lamina the moment of inertia is least for a set of parallel axes in its plane when the axis contains the centre of mass. Considering axes in the plane of the lamina passing through a fixed point \(P\), if the moment of inertia for three such axes has the same value, prove that the value is equal for all such axes. Show that a uniform triangular lamina of mass \(m\) has the same moment of inertia for any coplanar axis as the system of three masses each of \(\frac{1}{3}m\) at the mid-points of the sides.
A particle is attached to one end of a light perfectly flexible string of length \(a\) whose other end \(O\) is fixed. When hanging at rest the particle is given a horizontal velocity \(u\). Find conditions to ensure that \(O\) will be the lowest point at which, in the subsequent motion the string remains taut, and show that if these conditions are not satisfied the particle will pass through \(O\) if \(u^2 = (2 + \sqrt{3})ga\).
A particle is released from rest and slides under gravity down a rough rigid wire in the shape of a loop of a cycloid held fixed in a vertical plane with its line of cusps horizontal and uppermost. If the particle starts from a cusp and comes to rest at the lowest point, prove that the coefficient of friction \(\mu\) must satisfy the equation \(\mu^2 = e^{-\mu\pi}\). [The usual parametric equations for the cycloid may be taken in the form: $$x = a(\theta + \sin\theta), \quad y = a(1 - \cos\theta).]$$
A particle moving under gravity in a medium offering resistance proportional to the speed suffers an explosion in which it splits into two parts of equal masses, the speed being relative one to the other not necessarily in the same direction as the combined velocity of the undivided particle immediately before the explosion. Prove that at any subsequent instant the distance separating the two fragments is given by $$d = \frac{v}{k}(1 - e^{-kt}),$$ where \(v\) is the initial relative velocity and \(k\) is a constant.
A smooth rigid wire in the form of a parabola is held fixed in a vertical plane with its vertex downwards. A bead moves under gravity on the wire. Prove that at the square of the normal reaction of the bead on the wire is inversely proportional to the cube of the height of the point above the directrix of the parabola.
A particle \(P\) describes an orbit under a force per unit mass directed towards a fixed origin \(O\) of magnitude \(\mu r^{-2} + \lambda r^{-3}\), where \(r\) is the length \(OP\) and \(\mu\) and \(\lambda\) are constants. Prove that if \(\lambda\) is small enough the path in general can be expressed by the equation \(l = r(1 + e\cos n\theta)\), where \(l\), \(e\), and \(n\) are constants, and where \(\theta\) is measured from a suitable radius.
A uniform rough solid sphere is projected up a line of greatest slope of a plane inclined at an angle \(\alpha\) to the horizontal. Initially the velocity of the centre of the sphere is \(v\) and there is no angular velocity. Obtain an equation giving the variation of the velocity between the sphere and the plane at the point of contact for the motion immediately after the projection, and prove that the velocity of slip will vanish instantaneously after a time \(2\mu g(7\mu\cos\alpha + 2\sin\alpha)\), where \(\mu\) is the coefficient of friction between the sphere and the plane. Prove also that whatever the value of \(\mu\), the velocity of slip will either be zero or change in sign during the subsequent motion.
A thin rod of mass \(M\), not necessarily uniform, is suspended from one end \(O\) and can turn freely about \(O\) in a vertical plane. The rod is set in motion from its equilibrium position by a horizontal blow \(J\) delivered at some point \(X\) below \(O\). The impulsive reaction at \(O\) must not exceed a value \(H\) owing to the weakness of the hinge. Find the maximum value of the kinetic energy of the instantaneous motion of the rod, and prove that the depth below \(O\) of the point \(X\) in this case is given by $$OX = \left(1 + \frac{H}{J}\right)\frac{k^2}{h},$$ where \(k\) is the radius of gyration of the rod about \(O\), and \(h\) the depth below \(O\) of its centre of mass.