Two particles of masses \(m, m'\) connected by a light rod of length \(a+b\) are moving on a smooth horizontal plane in a direction perpendicular to the rod with uniform velocity \(v\); if the rod strikes a small fixed inelastic obstacle at a point whose distances from the masses are \(a, b\) respectively, prove that the measure of the blow is \(mm'(a+b)^2v/(ma^2+m'b^2)\).
A horizontal plate with a particle resting on it is made to oscillate vertically with simple harmonic motion through a distance of 2 feet. What is the smallest number of vibrations per minute that will throw the particle off the plate?
Find the direction and magnitude of the least force which will keep a weight \(W\) at rest on a rough incline of angle \(\alpha\). A uniform cube is supported on a rough incline with four of its edges horizontal by a string attached to the middle point of its highest edge and parallel to the direction of greatest slope of the incline. Prove that the greatest slope \(\alpha\) of the incline for equilibrium to be possible is given by \(\tan\alpha = 2\mu+1\), where \(\mu\) is the coefficient of friction.
\(AB\) and \(CD\) are light rods hinged at fixed points \(A\) and \(C\), and \(AC\) is equal to \(CD\), \(C\) being vertically below \(A\). To \(D\) is attached a heavy ring of weight \(W\) which can slide on \(AB\). If a force \(P\) applied at \(B\) at right angles to \(AB\) keeps the system at rest, show that \(\frac{P}{W}=\frac{2DN}{AB}\), where \(N\) is the foot of the perpendicular from \(D\) on \(AC\).
Seven equal light rods are smoothly jointed so as to form three equilateral triangles \(ABD, BDE, BEC\), the straight line \(ABC\) being horizontal and above \(DE\). Weights of 1 lb. are hung from \(A\) and \(C\) and weights of 2 lb. from \(D\) and \(E\), and the system is suspended from \(B\). Draw the force diagram.
An engine working at 600 H.P. pulls a train of 250 tons along a level track and the resistance is 16 lb. per ton. When the velocity of the train is 30 miles per hour, find its acceleration in foot-second units. At what uniform speed can the engine working at this H.P. pull the train up a slope of 1 in 100, with the same resistance as before?
Find the loss of kinetic energy when two elastic spherical balls collide directly. A small ball of mass \(m\) is just displaced from the top of a circular tube in a vertical plane and, falling down the tube, collides with a small ball of mass \(2m\) resting at the bottom. Find the height to which the latter ball rises after the second impact, in terms of \(a\), the radius of the tube, and \(e\) the coefficient of elasticity.
Define simple harmonic motion, and find an expression for the period in seconds if the retardation is \(f\) in foot-second units when the displacement is \(a\) feet. A mass \(m\) lb. is hung from the end of an elastic string so that in equilibrium the extension is \(a\) feet. If when in this position it is given a velocity \(v\) vertically downwards, show that the greatest extension (beyond the natural length) is \(v\sqrt{a/g}\) feet; and find the period, assuming the string to remain stretched throughout the oscillation.
Use Leibnitz's theorem to show that the \(n\)th differential coefficient of \(x^{n-1}\log x\) is \(\frac{C}{x}\) where \(C\) is a constant, and find the value of \(C\). If \(y=(x^2-1)^n\), show that \((x^2-1)\frac{dy}{dx}=2nxy\). Differentiate this equation \(n\) times and, writing \(z=\frac{d^ny}{dx^n}\) in the result, show that \((1-x^2)\frac{d^2z}{dx^2}-2x\frac{dz}{dx}+n(n+1)z=0\).
A window consists of a rectangular frame surmounted by a semicircle. If the perimeter of the window is given, prove that the area is a maximum when the diameter of the semicircle (the width of the window) is twice the height of the rectangle. Find the maximum area when the given perimeter is 24 feet.