The light pin-jointed framework shown in the figure is supported freely at points \(A\) and \(B\) at the same level. \(AC=CD=DE=EB=a\); \(CF\) and \(EG\) are each vertical and equal to \(\frac{1}{2}a\). Determine the force set up in the member \(FD\) when a load \(W\) is placed successively at \(C, D\) and \(E\). Show that, as the load moves slowly along \(CD\), the force in \(FD\) varies linearly with the distance of the load from \(A\), and hence find the point in \(CD\) at which the load will produce no force in \(FD\). \centerline{\includegraphics{c1955-q1-mechanics-1.png}}
A uniform rigid plank, of length \(l\) and weight \(W\), when laid on soft ground sinks uniformly through a distance \(d\) (small compared with \(l\)). A concentrated load is then placed on the plank and it is found that the depressions at the ends are now \(1.5d\) and \(4.5d\). Find the magnitude and position of the added load assuming that the ground exerts a vertical pressure on the plank, per unit length, which at every point is proportional to the depression produced there.
A gun of mass \(M\), which can recoil freely on a horizontal platform, fires a shell of mass \(m\), the elevation of the gun being \(\alpha\). Show that the angle \(\phi\) which the path of the shell initially makes with the horizontal is given by \(\tan\phi = (1+\frac{m}{M})\tan\alpha\). Assuming that the whole energy of the explosion is transferred to the shell and the gun, show that the energy given to the shell is less than it would be if the gun were fixed, in the ratio \(M:(M+m \cos^2\phi)\).
Two light rods \(AB\) and \(BC\) are hinged together at \(B\); \(BC\) turns on a hinge at a fixed point \(C\), and the end \(A\) of \(AB\) is constrained to move in a smooth guide along \(AC\), the axes of the hinges being at right angles to the plane \(ABC\). The hinges at \(B\) and \(C\) are tightened so that relative rotation is resisted by friction couples \(L\) and \(M\) respectively, and a force \(F\) is applied at \(A\) along \(AC\). Show that motion will not occur unless \(F\) exceeds the value \[ \frac{L}{BN} + \frac{M\cdot AN}{AC \cdot BN}, \] where \(N\) is the foot of the perpendicular from \(B\) on to \(AC\).
A drum, of radius \(a\) and moment of inertia \(I\) about its axis, is free to rotate about a horizontal axial spindle. A heavy chain of length \(l\) and mass \(ml\) is placed across the drum in a plane at right angles to the axis so that equal lengths hang freely on both sides. The equilibrium is then slightly disturbed. Obtain an expression for the velocity of the chain when one end has fallen a distance \(x\), assuming that the chain does not slip on the drum and that it is still in contact with it over half the circumference. Obtain also an expression for the acceleration of the vertical portions of the chain at this instant.
A thin uniform rod rests at one end on a horizontal plane while the other end is slowly raised by means of a wedge forced under it, the axis of the rod being coplanar with a line of greatest slope of the wedge. The angle of the wedge is 45\(^\circ\) and the coefficient of friction at the points of contact of the rod with the plane and with the wedge is \(\frac{1}{2}\). Show that until the rod reaches an angle \(\tan^{-1}\frac{1}{2}\) to the horizontal an additional force will be required at the lower end to prevent it slipping.
Obtain an expression for the energy required to raise a mass \(m\), initially at rest on the surface of the earth, to a height \(h\) above it with a final velocity sufficient to enable it to circle the earth continuously at that height, neglecting the effects of air resistance. (The acceleration due to gravity at a height \(x\) may be taken as \((\frac{r}{r+x})^2 g\) where \(r\) is the radius of the earth.)
A rigid uniform plank \(ABC\) of mass 30 lb. can turn freely about a fixed horizontal hinge at \(B\) and rests on a support at \(A\) so that \(ABC\) is horizontal. \(AB=BC=3\) ft. A concentrated mass of 30 lb. is attached to the end \(A\). To what height will the end \(A\) rise if a concentrated mass of 20 lb. is dropped from a height of 1 ft. on to the end \(C\) and adheres to it without rebound?
An engine is coupled to a flywheel of mass 100 lb. and radius of gyration 2 feet. At a particular instant it is turning at a speed of 2 revolutions per second and during the next revolution it is subjected to a net torque given by \(16,000 \cos 2\theta\) foot-poundals, where \(\theta\) is the angle turned through in radians. Find approximately the maximum and minimum speeds in revs. per sec. during the motion.
Two light elastic strings, \(AB\) and \(CD\), of the same unstretched length but of different elasticity, extend by amounts \(a\) and \(b\) respectively when a certain mass hangs in equilibrium on each in turn. Obtain expressions for the frequency of a vertical oscillation of the mass in each of the following cases: