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.
If \(y=a+x\sin y\), prove that when \(x=0\), \[ \frac{dy}{dx}=\sin a, \quad \text{and} \quad \frac{d^2y}{dx^2}=\sin 2a. \] Show by expanding \(e^{\frac{1}{x}}\) in descending powers of \(x\), that \[ \frac{d^n}{dx^n}(x^{n-1}e^{\frac{1}{x}}) = (-1)^n x^{-n-1}e^{\frac{1}{x}}. \] Is this procedure strictly legitimate?
Prove that the cone of greatest volume which can be inscribed in a given sphere has an altitude equal to \(\frac{2}{3}\) the diameter of the sphere.
Prove the formula \[ \rho = \frac{\{1+\left(\frac{dy}{dx}\right)^2\}^{\frac{3}{2}}}{\frac{d^2y}{dx^2}} \] for the radius of curvature of a curve. For the cycloid \[ x=a(\theta+\sin\theta), \quad y=a(1-\cos\theta), \] show that at a point defined by the angle \(\theta\), \(\frac{dy}{dx}=\tan\frac{1}{2}\theta\), and \(\rho=4a\cos\frac{1}{2}\theta\).
Prove that \(\tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right)\). Solve the equation \(\cot^{-1}x - \cot^{-1}(x+2) = \frac{\pi}{12}\).