Three unequal uniform rods \(AB, BC\) and \(CD\), of lengths \(a, b\) and \(c\) respectively, are smoothly jointed together at \(B\) and \(C\); the ends \(A\) and \(D\) slide smoothly along a horizontal rail. Find the positions of equilibrium, and illustrate by sketches the various cases.
A regular hexagonal framework \(ABCDEF\) is formed from six equal uniform rods, each of weight \(W\), smoothly jointed together; it is kept in shape by three light rods \(BE, BF\) and \(CE\). Find the thrust or tension in each of these three rods if the framework is suspended from \(A\).
A motor-car stands on level ground with its back wheels, which are of radius \(a\), in contact with a fixed obstacle of rectangular cross-section and height \(\frac{1}{2}a\). The coefficient of friction between the wheels and the ground and between the wheels and the obstacle is \(\mu\). These back wheels together carry a vertical load \(V\), including their own weight. A torque of gradually increasing moment \(M\) is applied to the back axle. Find how, and for what value of \(M\), equilibrium is broken, distinguishing the cases that can arise. Neglect the friction in the bearings.
\(ABC\) is a triangular lamina. Forces of magnitude \(k \cdot AB\) and \(k \cdot BC\) act outwards along the perpendicular bisectors of the two edges \(AB\) and \(BC\) respectively. Show that their resultant is a force of magnitude \(k \cdot AC\) acting inwards along the perpendicular bisector of the third edge \(AC\). State and prove a more general theorem about the resultant of forces acting along the perpendicular bisectors of the edges \(AB, BC, \dots, MN\) of a lamina in the form of a polygon \(ABC\dots MN\). Find the magnitude and line of action of the resultant of forces of magnitude \(k \cdot AB, k \cdot BC, \dots, k \cdot MN\) acting at the mid-points of the edges \(AB, BC, \dots, MN\) respectively of a lamina \(ABC\dots MN\) along lines (in the plane of the lamina) making the same angle \(\alpha\) with the corresponding edges.
A light inextensible string \(ABC\) is laid upon a smooth horizontal table with \(AB\) and \(BC\) straight and \(\angle ABC\) equal to \(135^\circ\). Equal particles are attached at \(A\) and \(B\), and the end \(C\) is then jerked into motion in the direction \(BC\) with velocity \(V\). Calculate the speed with which \(A\) is jerked into motion.
A particle is attached to a point \(P\) of a light uniform elastic string \(AB\). The ends of the string are fixed to points in a vertical line, and the particle oscillates along this line. Show that, provided that the two parts of the string remain taut, the motion is simple harmonic. Show also that the period is the same whatever the distance between the points to which \(A\) and \(B\) are attached and whichever end of the string is uppermost.
A uniform hemisphere of mass \(M\) and radius \(a\) rests with its plane face upon a smooth horizontal table. A particle of mass \(m\) falling vertically strikes the hemisphere and rebounds horizontally. After the impact the hemisphere slides along the table. If the coefficient of restitution between the particle and the hemisphere is \(e\) and the effect of friction is neglected, find the height above the table at which the hemisphere is struck.
A particle is released from rest on the surface of a smooth fixed sphere at a point whose angular distance from the highest point is \(\alpha\). Find the point where the particle leaves the surface, and prove that the angular distance of this point from the highest point of the sphere cannot be less than about \(48^\circ\). Show that the radius of curvature of the trajectory of the particle after it leaves the surface is initially equal to the radius of the sphere.
A rigid body rotates without friction about a fixed horizontal axis; the radius of gyration about the axis is \(k\) and the distance of the centre of gravity from the axis is \(h\). Prove that the angular displacement of the body from its position of stable equilibrium varies with time in the same way as that of a simple pendulum, of length \(k^2/h\), set in motion in a corresponding way. A rod \(AB\), not necessarily uniform, is of length \(l\). When it swings about the end \(A\) in a vertical plane the length of the equivalent simple pendulum is \(a\), and when it swings about the other end \(B\) in a vertical plane the length of the equivalent simple pendulum is \(b\). Find the distance of the centre of gravity from \(A\).
A uniform chain passes over a small smooth peg fixed at a height \(h\) above the edge of a table. From one side of the peg a length \(h+x\) of the chain hangs clear of the table. From the other side of the peg a length \(h\) hangs vertically, and the rest of the chain is heaped on the table below the peg. Assuming that the links are jerked into motion one by one, find a differential equation for \(x\) as a function of the time.