A smooth wire has the shape of a parabola whose latus rectum is of length \(l_0\) and whose axis is vertical and vertex upwards. Two beads \(A\) and \(B\), whose masses are \(m_1\) and \(m_2\), where \(0 < m_1 < m_2\), slide on the wire, and are joined by a light inelastic string of length \(l\), where \(l > 2a\), which passes through a small smooth ring at the focus of the parabola. Prove that the only positions of equilibrium for which the two beads are not on the same side of the vertex are those in which at least one of the beads is at the vertex of the parabola, and determine which positions are stable and which are unstable. How is the problem altered if \(m_1 = m_2\)?
A plane is inclined at an angle \(\alpha\) to the horizontal. Its surface is rough, but not uniformly rough, the coefficient of friction \(\mu\) being proportional to the distance \(r\) from a point \(O\) in the plane, \(\mu = kr\). A particle of mass \(m\) is placed on the plane at \(O\) and released from rest. How far does the particle travel before it comes to rest, and how long is it in motion before it comes to rest? Verify that the work wasted through friction is equal to the potential energy lost.
A car of mass \(m\) moves in a straight line on a level road. It is acted on by a constant propulsive force \(kv^2\), and the motion is opposed by a resisting force \(kv^2\) when the speed is \(v\). Prove that the steady speed at which the car can travel is \(c\), and that if it starts from rest it attains the speed \(v\) when it has travelled a distance \[ \frac{m}{2k} \log \frac{c^2}{c^2-v^2}. \] If the mass of the car is one ton, the steady speed is 60 miles per hour, and the horse-power developed by the engine at this speed is 30, find, correct to the nearest foot, the distance travelled when the car attains a speed of 30 miles per hour. [Assume \(g = 32\) ft. sec.\(^{-2}\).]
A bead of mass \(m\) slides on a smooth wire in the form of a circle of radius \(a\) which is fixed in a vertical plane. The bead is projected from the lowest point of the circle at the instant \(t = 0\) with velocity \(2\sqrt{(ga)}\), and in the subsequent motion the radius from the centre of the circle to the bead makes an angle \(\theta\) with the downward vertical at time \(t\). Prove that \[ \sin \frac{\theta}{2} = \tanh nt, \] where \(n^2 = g/a\). If \(R\) is the reaction of the wire on the bead at any time during the motion, \(R\) being measured towards the centre of the circle, express \(R\) (i) as a function of \(\theta\), and (ii) as a function of \(t\).
The maximum range of a certain gun on a horizontal plane is \(2h\). The gun is placed at the highest point of a hill in the form of a hemisphere of radius \(a\), where \(a > h\). Prove that the area of the part of the surface of the hill which is commanded by the gun is \[ \pi a \{a - \sqrt{(a-4h)}\}^2. \] Examine the limit to which this expression tends as \(a\) tends to infinity.
One point \(O\) of a rigid lamina of mass \(M\) is fixed, and the lamina is free to swing about \(O\), without friction, in a vertical plane. If the lamina executes small oscillations about the position of stable equilibrium, prove that the length of the equivalent simple pendulum is \[ h + \frac{l^2}{h}, \] where \(h\) is the distance of the centre of gravity \(G\) from \(O\), and \(Ml^2\) is the moment of inertia of the lamina about \(G\). If the lamina has uniform surface density, and has the form of an annulus bounded by concentric circles of radii \(a\) and \(b\), where \(a < b\), and if the point of suspension \(O\) can have any position in the annulus, prove that the least possible value for the length of the equivalent simple pendulum is \(\sqrt{2(a^2 + b^2)}\).
Weights \(P\) and \(Q\) are attached to the ends of a light flexible rope which is in limiting equilibrium hanging over a rough circular cylinder, the rope lying in a plane perpendicular to the axis of the cylinder which is horizontal. If \(Q\) is on the point of ascending, what weight must be added to it so that it becomes on the point of descending?
A body consists of a uniform solid hemisphere of radius \(a\) and a uniform solid right circular cone of base radius \(a\) and height \(h\) of the same density as the hemisphere, the base of the cone coinciding with the circular face of the hemisphere. Find the greatest permissible value of \(h/a\) in order that the body may be in stable equilibrium in an upright position with the hemisphere resting on a horizontal table.
A uniform rod \(AB\) of length \(l\) lies in a horizontal position on a rough inclined plane of angle \(\alpha\) for which the coefficient of friction \(\mu > \tan \alpha\). At the end \(B\) a gradually increasing force is applied acting upwards along the line of greatest slope. If the rod starts to turn about a point \(O\) such that \(OB = p\), show that \[(p/l)^2 = (\mu - \tan \alpha)/2\mu,\] and hence show that the length of rod that begins to move upwards is less than \(l/\sqrt{2}\).
A flat plate of uniform thin material is in the form of a plane quadrilateral \(ABCD\). The diagonals meet at a point \(O\). Show that its centre of mass coincides with that of four particles each of mass \(m\) at \(A\), \(B\), \(C\), \(D\) and one of mass \(-m\) at \(O\).