A uniform rigid rod \(AB\) weighing 12 lb. is hung from a rigid horizontal beam by three equal vertical wires, one at each end and one at the middle point. A weight of 18 lb. is attached to the rod at \(C\), where \(AC=\frac{1}{4}AB\). If the wires obey Hooke's law, find the pull in each wire.
Seven equal uniform rods \(AB, BC, CD, DE, EF, FG, GA\), are freely jointed at their extremities and rest in a vertical plane supported by small light rings at \(A\) and \(C\), which can slide on a smooth fixed horizontal rod. If \(\theta, \phi, \psi\) are the angles that \(BA, AG, GF\), make with the vertical, prove that \[ \tan\theta = 4\tan\phi = 2\tan\psi. \]
A thin uniform straight rod \(PQ\) of weight \(W\) rests partly within and partly without a uniform cylindrical jar of weight \(4W\), which stands on a horizontal table. The rod rests in contact with the smooth rim of the jar, with its end \(P\) pressing against the rough curved surface of the jar. If the rod is about to slip and the jar is about to upset simultaneously, prove that the rod makes with the vertical an angle \[ \frac{1}{2}\lambda + \frac{1}{4}\cos^{-1}\left(\frac{1}{3}\cos\lambda\right), \] where \(\lambda\) is the angle of friction.
A ball whose coefficient of restitution is \(e\) is projected with velocity \(v\) at an inclination \(\alpha\) to the horizontal from a point \(A\) on a horizontal plane. \(A\) is at a distance \(d\) from a vertical wall. The ball strikes the wall, and then after rebounding once on the horizontal plane returns to \(A\). Prove that \[ v^2e\sin2\alpha=gd. \]
Two particles, each of mass \(m\), are attached to the ends of a long fine inextensible string, which hangs over two small smooth pegs which are at the same level and \(2a\) apart. A particle of mass \(2m\) is attached to the string midway between the pegs and is then let go. Prove that during the subsequent motion, if \(\phi\) is the angle between the two non-parallel parts of the string, the velocity of the mass \(2m\) is \[ 2\sqrt{ag\frac{1-\tan\frac{\phi}{4}}{3+\cos\phi}}. \]
A uniform rectangular plate \(ABCD\) is hinged at the fixed point \(A\) and is supported in such a position that \(AB\), one of the longer sides, is horizontal, and \(AD\) is vertical. When the plate is released it swings in its own plane about the fixed hinge \(A\) and comes to rest with \(AB\) vertical. The stiffness of the hinge produces a constant retarding couple during motion. Prove that the plate stays in the new position if \[ \frac{AB}{AD} > 1+\frac{\pi}{2}. \]
The acceleration of a certain racing motor car at a speed of \(v\) feet per second is \(\left(3.6 - \frac{v^2}{9000}\right)\) feet per second per second. Find the maximum speed of the car, and prove that from a standing start a speed of 150 feet per second is acquired in one minute after travelling 1800 yards. Assume that \(\log_e 6=1.8\), and \(\log_e 11 = 2.4\).
If \[ y = \frac{\sin^{-1}x}{\sqrt{1-x^2}}, \] prove that \[ (1-x^2)\frac{dy}{dx} = xy+1; \] and if \(y_n\) denotes the \(n\)th differential coefficient of \(y\), prove that, when \(x=0\), \[ y_n=(n-1)^2y_{n-2}. \] Prove that the limit of \((\cos x)^{\cot^2x}\) as \(x\) tends to zero is \(\displaystyle\frac{1}{\sqrt{e}}\).
If \(\alpha\) and \(\beta\) are given acute angles, and \(\alpha>\beta\), prove that the maximum and minimum values of \[ \frac{1+2x\cos\alpha+x^2}{1+2x\cos\beta+x^2} \] are \[ \frac{1-\cos\alpha}{1-\cos\beta} \quad \text{and} \quad \frac{1+\cos\alpha}{1+\cos\beta} \text{ respectively}. \]
Sketch the locus of a point \(P\) for which \[ x=a\cos^3\phi, \quad y=a\sin^3\phi, \] where \(a\) is constant and \(\phi\) is variable. Prove that the tangent at \(P\) to the locus is \[ x\sec\phi+y\csc\phi=a, \] and that the whole length of the curve is \(6a\).