Two particles of masses \(4m, 3m\) connected by a taut light string of length \(\frac{1}{2}\pi a\) rest in equilibrium on a smooth horizontal cylinder of radius \(a\). If equilibrium is slightly disturbed so that the heavier particle begins to descend, find at what point it will leave the surface, and shew that at that moment the pressure on the other particle is slightly greater than two-thirds of its weight. [Trigonometrical tables should be used.]
One end of a light inelastic string is attached to a fixed point A of a rod, which is held at an inclination \(\alpha\) to the horizontal, and the other end of the string is attached to a small smooth ring C, of weight \(w\), which slides on the rod. A similar ring B slides on the string. If \(\alpha < \pi/6\), prove that in equilibrium each part of the string makes with the vertical an angle \(\beta\), where \(\tan\beta = 3\tan\alpha\).
A number of coplanar forces act at various points of a rigid body. Prove that, if the vector sum of the forces is zero, the system is equivalent to a couple; and that if, in addition, the moment of the system about one point of the plane is zero, then the system is in equilibrium. Forces proportional to the sides act at the mid-points of the sides of a simple polygon, the forces being perpendicular to the sides and directed inwards. Prove that the system is in equilibrium.
A region of a plane, bounded by a simple closed curve, is rotated about a line in the plane; the line does not intersect the region. Prove that the volume of the ring-shaped solid so formed is the product of the area of the region and the length of the path of its centroid. A semicircular area, of radius \(a\), is rotated about a line in its plane parallel to, and at a distance \(b\) from, the bounding diameter: the axis of rotation is on the side of the diameter remote from the semicircle. Prove that the volume of the solid so formed is \(\frac{1}{2}\pi a^2(3\pi b+4a)\). Also determine the area of the surface of the solid.
A uniform heavy inelastic string hangs over a circular cylinder of radius \(a\) which is fixed with its axis horizontal. The string lies in a plane perpendicular to the axis of the cylinder, and the lengths of the straight pieces of the string hanging on either side are \(l\) and \(l'\), where \(l > l'\). The coefficient of friction is unity, and the string is on the point of slipping. Prove that \(l' = a+(a+l)e^\pi\).
Find the form in which a uniform heavy inelastic string hangs under gravity. The ends A, B of a uniform string, of length \(2b\) and weight \(W\), are held at the same level, and the sag in the middle is \(h\). Prove that the tension at either end is \[ \frac{b^2+h^2}{4bh} W. \] Prove also that the distance from A to B is \[ \frac{b^2-h^2}{h}\log\frac{b+h}{b-h}. \]
A particle is attached to the mid-point of a light elastic string of natural length \(a\). The ends of the string are attached to fixed points A and B, A being at a height \(2a\) vertically above B, and in equilibrium the particle rests at a depth \(5a/4\) below A. The particle is projected vertically downwards from this position with velocity \(\sqrt{(ga)}\). Prove that the lower string slackens after a time \(\frac{\pi}{12}\sqrt{\frac{a}{g}}\), and that the particle comes to rest after a further time \(\beta \sqrt{\frac{a}{2g}}\), where \(\beta\) is the acute angle defined by the equation \(\tan\beta=\sqrt{3}\).
A truck runs down an incline of 1 in 100; the resistance to motion is proportional to the square of the speed, and the terminal velocity is 40 miles per hour. Prove that the truck, starting from rest, acquires a velocity of 20 miles per hour in a distance of about 1550 feet.
Prove that the path of a particle under gravity is a parabola whose directrix is the energy level (i.e. the total energy is equal to the potential energy that the particle would have if placed at a point on the directrix). It is required to throw a ball from a point A to a point B. Prove that, if the velocity of projection is a minimum, the focus of the parabolic path lies on AB, and that the velocity of projection is \(\sqrt{\{g(l+h)\}}\), where \(l\) is the length AB, and \(h\) is the height of B above A. Prove also that, for any velocity of projection greater than the minimum, there are two paths, whose foci are equidistant from the line AB.
One end of a long light inelastic thread is attached to a point on the surface of a smooth circular cylinder, of radius \(a\). The cylinder can rotate about its axis, which is horizontal. The thread is wound on the cylinder, and a particle of mass \(M\) hangs from the free end. The system starts from rest. The moment of inertia of the cylinder about its axis is \(Mk^2\), and its rotation is opposed by a constant frictional couple \(Mgb\), where \(b