A particle is projected with a given velocity from a point \(P\) to pass through another given point \(Q\) at horizontal and vertical distances \(a\) and \(b\) respectively from \(P\). Prove that if \(H\) is the difference in the greatest heights and \(R\) the difference in the ranges on the horizontal plane through \(P\) for the two possible paths, then \(H/R = \frac{1}{2}a/b\).
Three particles of masses \(m, m', m''\) are attached to the points \(A, B, C\) of an inextensible string. They are laid on a smooth horizontal table, with the portions of the string between the particles taut and the angle \(ABC\) obtuse, and equal to \(\pi-\alpha\). A blow \(P\) is applied to \(C\) parallel to and in the direction \(AB\). Prove that \(m\) begins to move with velocity \(m'P \cos^2\alpha/[m'(m+m'+m'') + mm'' \sin^2\alpha]\).
A smooth wedge of mass \(M\) and angle \(\alpha\) rests on a smooth horizontal plane on which it is free to slide. Another wedge of the same mass and angle is placed on the face of the first, so that the upper face of the second is horizontal. A smooth particle of mass \(m\) is placed on the upper face of the second, and the whole system is allowed to move freely under gravity. Prove that, if the two wedges can only move in directions perpendicular to their edges, \(m\) descends vertically with acceleration \[ \frac{2(m+M)\sin^2\alpha}{M+(2m+M)\sin^2\alpha}g. \]
Prove that if three forces acting upon a rigid body are in equilibrium, their lines of action must all lie in the same plane, and must either pass through a point or be parallel. A uniform rod of length \(2a\) and weight \(W\) hangs in an oblique position, supported by an inextensible string of length \(2l\) (\(l>a\)) whose ends are fastened to the ends of the rod and which passes over a smooth peg, and a weight \(w\) is attached to the rod at a distance \(d\) from its middle point. Prove that the lengths of the string on the two sides of the peg are \[ l\left(1-\frac{d}{ak}\right), \quad l\left(1+\frac{d}{ak}\right), \] where \[ k=1+\frac{W}{w}. \]
A block of stone of weight \(W\) is placed on a rough plane whose inclination \(\alpha\) to the horizontal is less than \(\epsilon\) the angle of friction. Prove that the least force that will drag it up the plane is \[ W\sin(\alpha+\epsilon). \] A uniform rectangular board is supported with its plane vertical and with two edges, of length \(a\), horizontal, by pressures applied at two points one in each of its vertical edges, at which the coefficient of friction is \(\mu\). Prove that the vertical distance between the points of support cannot exceed \(\mu a\).
Six equal heavy beams are freely jointed at their ends to form a hexagon, and are placed in a vertical plane with one beam resting on a horizontal plane; the middle points of the two upper slant beams, which are inclined at an angle \(\theta\) to the horizon, are connected by a light cord. Show that its tension is \(6W\cot\theta\), where \(W\) is the weight of each beam.
Some cubical blocks of stone are resting on a breakwater when it is swept by a heavy sea. The velocity of the wave, which impinges directly on the face of the block, is estimated at 30 feet per second. If the blocks weigh 120 pounds per cubic foot, and sea water weighs 64 pounds per cubic foot, show that the impact of the water will be sufficient to overturn a block weighing as much as 180 tons.
Prove that the envelope of the paths of particles projected in vacuo from the same point, with the same velocity, and in the same vertical plane, is a parabola with the point of projection as focus. A parabolic trajectory passes through two given points, in directions inclined at angles \(\alpha, \beta\) to the horizontal. Show that for all such trajectories \((\tan\alpha+\tan\beta)\) has the same value.
A particle moves along the smooth interior of a straight tube which itself is moving in the direction of its own length on a smooth horizontal table, both ends of the tube being closed. If the coefficient of restitution is \(e\), show that the energy lost in consecutive impacts diminishes in the ratio \(e^2:1\), and that the time between consecutive impacts increases in the ratio \(1:e\).
Define simple harmonic motion, and show that the velocity at any displacement \(x\) from the centre of motion is \(\frac{2\pi}{p}\sqrt{a^2-x^2}\), where \(p\) is the period and \(a\) the amplitude of the oscillation. A mass of 10 lb. is hung at the end of a spring which requires 5 lb. to stretch it one inch. Find the period when the mass is set in oscillation, and find the maximum kinetic energy of the mass when the amplitude is 3 inches.