A ladder standing on smooth ground rests with its upper end against a smooth vertical wall. Prove that it can be held in a given position by a cord fastened to a point \(O\) in the base of the wall and attached to any point \(P\) on a certain part of the ladder; and that for different positions of \(P\) the tension of the cord varies as \(\text{cosec } \angle COP\), where \(C\) is the middle point of the ladder.
Prove that three forces in equilibrium must be co-planar and meet in a point or be parallel. A wire fence turns through a right angle at a corner. The wires pass through holes in the posts, and each wire has the same tension throughout. Shew that the corner post may be supported without pressure on its lower end by two bars, one in tension, the other in thrust; the bars being attached to the post at two points and having their other ends fixed to the ground.
Explain how to construct a funicular polygon for forces in one plane, and prove that for equilibrium both the force and the funicular polygons must be closed. A straight rod is acted on by given vertical forces at given points, and supported at two other given points. Supposing that the reactions at these points are vertical, shew how to determine their magnitudes by means of a funicular polygon.
Two cylinders lie in equilibrium on a rough inclined plane, with their axes horizontal and in contact with one another. The upper cylinder, of radius \(a\), is heavy but the lower cylinder, of radius \(b\), is of negligible weight. Prove that, if \(\alpha\) is the inclination of the plane to the horizontal, \(b > a \tan^2 \alpha\); that the coefficient of friction between the heavy cylinder and the plane must be at least \[ \frac{1}{2 \cot\alpha - \sqrt{a/b}}; \] and that the other coefficients of friction must be at least equal to \(\sqrt{b/a}\).
A rod \(OA\) revolves in one plane about \(O\) as a fixed point with constant angular velocity \(n\), and a second rod \(AB\) revolves in the same plane with constant angular velocity \(n'\). Prove that, if \(C, D\) are certain points fixed in \(OA\), \(CB\) is normal to the path of \(B\) and \(BD\) represents the acceleration of \(B\). Prove, by dynamical considerations, that the radius of curvature of the path of \(B\) is equal to \(BC^2/BE\), where \(E\) is the foot of the perpendicular from \(D\) on \(BC\).
Explain what is meant by the dimensions of a physical quantity, and illustrate the explanation by comparing the dynamical units of work in the ft. lb. sec. and C.G.S. systems, taking 1 ft. = 30 cm. and 1 lb. = 450 gr. The driving wheels of a locomotive exert a constant force on the rails when the velocity \(\le V\) ft./sec., and the engine works at constant horse power \(H\) when the velocity \(\ge V\) ft./sec. If the train, starting from rest, attains the velocity \(V\) in \(t\) seconds, prove that full speed is \[ \frac{550HVgt}{550Hgt - \frac{1}{2}MV^2} \text{ ft./sec.,} \] where \(M\) lbs. is the mass of the train (including engine) and the frictional resistance to motion is supposed to be constant.
Particles are projected from a given point \(A\) in different directions with a given speed \(V\). Find the envelope of the trajectories. Prove that to obtain the greatest range on a horizontal plane at depth \(h\) below \(A\) the direction of projection must make with the horizontal an angle \[ \cos^{-1}\sqrt{\frac{gh}{V^2+gh}}. \]
Two balls impinge directly. Find the amount of momentum transferred from one to the other. Two equal balls, with coefficient of restitution equal to \(\frac{2}{3}\), impinge directly with relative velocity 1 ft./sec. Assuming that contact lasts for \(\cdot 0002\) sec., and that during this time the force on either ball increases uniformly with the time to a maximum and then decreases uniformly to zero, compare the maximum force with the weight of a ball.
A particle tied to a fixed point \(O\) by an inextensible string of length \(a\) is projected horizontally from the lowest position so that the string becomes slack when the particle is at height \(h\) above \(O\). Find the velocity of projection, and prove that the string will again become taut after an additional time \[ 4\sqrt{\frac{h}{g}\left(1 - \frac{h^2}{a^2}\right)}. \]
A particle performs harmonic oscillations of amplitude \(a\) in a period \(T\). Find the velocity of the particle when at distance \(x\) from the position of equilibrium.
A particle \(P\) suspended by an extensible string \(AP\) is in equilibrium, and the extension of the string is equal to \(c\). Suddenly \(A\) is raised vertically through a height \(b(