(a) \(ABCO\) is a quadrilateral in which \(AB=BC\), \(CO=OA\), and the lengths of the sides are given. Given forces act along \(AB\) and \(BC\). Show that the moment about \(O\) is a maximum when the points \(A, B, C, O\) are concyclic. \item[(b)] Four forces act, in the sense \(\vec{AB}, \vec{BC}, \vec{CD}, \vec{DA}\), along the sides of a quadrilateral \(ABCD\) inscribed in a circle. If each force is inversely proportional to the length of the side along which it acts, show that the resultant force passes through the points of intersection of \(AB, CD\) and of \(AD, BC\).
A rod \(AB\) can pivot freely about the end \(A\), which is fixed, and is in equilibrium with the end \(B\) resting against the vertical face of a rough wall. \(O\) is the foot of the perpendicular from \(A\) on to the face of the wall, and \(OA=OB\). Find, in terms of the coefficient of friction between the rod and the wall, the angle which \(OB\) makes with the vertical direction when the rod is about to slip.
A heavy uniform chain of length \(2l\) (\(l>\pi a\)) hangs in equilibrium in a closed loop over a smooth circular pulley of radius \(a\) which is fixed in a vertical plane. If \(2\theta\) is the angle subtended at the centre of the pulley by the arc of that part of the chain in contact with the pulley, show that \[ \sinh\left(\frac{a\sin\theta\tan\theta}{l-a\theta}\right) = \tan\theta. \]
A uniform rigid wire \(ABC\) consisting of a straight section \(AB\) of length \(2l\) at right angles to a straight section \(BC\) of length \(4l\) is freely suspended at \(A\). Show that in the position of stable equilibrium \(AB\) makes an angle \(\tan^{-1}4/5\) with the downward vertical. If the wire makes small oscillations in the vertical plane about the position of equilibrium, find the length of the equivalent simple pendulum.
A small perfectly elastic sphere is projected with speed \(v\) from a point \(O\) on level ground towards the vertical face of a smooth wall of height \(h\). If \(O\) is at a distance \(d<2h\) from the face of the wall, show that there are two directions of projection for which the sphere first strikes the ground at \(O\) after bouncing off the wall provided that \[ g(d^2+4h^2)/(2h) > v^2 > 2gd. \]
A uniform circular disc of mass \(6m\) can rotate freely in a vertical plane about its centre \(O\), which is fixed. A particle of mass \(m\) is rigidly attached to a point on the rim of the disc, and initially the system is in equilibrium with the particle beneath \(O\). A horizontal impulse \(I\) is applied in the plane of the disc at its highest point. Find the impulsive reaction at \(O\). What is the condition on \(I\) for the disc to make complete revolutions?
Two uniform perfectly elastic smooth spheres, each of mass \(m\) and radius \(a\), are at rest on a horizontal table with their centres a distance \(4b\) apart (\(b < a\)). A third identical sphere rolls on the table towards them with velocity \(u\) in a direction normal to their line of centres and strikes them simultaneously. Find the velocities of the three spheres after the impact. Explain briefly why the results would be different if the conditions of the problem were altered very slightly in such a way that the impacts between the rolling sphere and the stationary spheres were not quite simultaneous.
A cube of wood of side \(a\) and mass \(M\) is initially at rest on a smooth horizontal platform. A bullet of mass \(m\) strikes the mid-point of a vertical face of the cube when travelling normal to the face with velocity \(u\). If the resistance of the wood to the bullet is \(\lambda v^{1/2}\), where \(v\) is the velocity of the bullet relative to the cube, show that the bullet passes right through the cube if \(u > \left(\frac{3m+M}{2mM}\lambda a\right)^{2/3}\), and that the time it spends in the cube is then \[ \frac{2mM}{\lambda(m+M)}\left\{u^{1/2} - \left(u^{3/2} - \frac{3m+M}{2mM}\lambda a \right)^{1/3}\right\}. \] [The effect of gravity is to be neglected.]
A uniform solid circular cylinder of mass \(2m\) and radius \(a\) can rotate freely about its axis which is fixed in a horizontal position. A particle of mass \(m\), initially at rest on the top of the rough surface of the cylinder, is slightly displaced. If the coefficient of friction between the particle and the cylinder is \(\frac{1}{2}\), show that the particle begins to slip after the cylinder has rotated through an angle \(\tan^{-1}\frac{1}{2}\).
A smooth hollow right circular cone of semi-angle 45\(^\circ\) is fixed with its axis vertical and its vertex \(O\) pointing downwards. A light elastic string of natural length \(a\) and modulus \(\lambda mg\) passes through a small hole in the cone at \(O\). One end of the string is fixed at a point distant \(a\) vertically below \(O\) and the other end is attached to a particle of mass \(m\) which travels on the inner surface of the cone. Initially the particle is projected horizontally with velocity \(4\sqrt{(ag/3)}\) in a direction tangential to the surface of the cone at a vertical height \(a\) above \(O\). If, in the subsequent motion, \(v\) is the component of velocity of the particle along a generator of the cone and \(y\) is the height of the particle above \(O\), show from the equations of energy and angular momentum that \[ \frac{v^2}{2g} = 4a - \frac{8a^3}{3y^2} - y - \frac{y^2}{3a}. \] Deduce that the height of the particle above \(O\) is always between \(a\) and \(2a\).