A uniform rod rests with its ends on two smooth planes inclined at \(30^\circ\) and \(45^\circ\) respectively to the horizontal. Prove that the inclination of the rod will be \(\cot^{-1}(\sqrt{3}+1)\). Also find what weight, fixed to the rod at a quarter of its length from one end, will suffice to enable the rod to rest horizontally.
Three equal spheres are lying in contact on a horizontal plane and are held together by a string which passes round them. A tube of weight \(W\) is placed with one diagonal vertical so that its lower faces touch the spheres, and the cube is supported in this position by the spheres. Show that the tension in the string is \(\frac{1}{3\sqrt{6}}W\), all friction being neglected. Note: The scanned document contains `1/3 sqrt(6) W`, which is ambiguous. Transcribed as the most likely mathematical meaning.
Explain the difference between stable, unstable and neutral equilibrium. A heavy flexible chain of weight \(w\) per unit length hangs over a light circular pulley of radius \(a\), the free parts of the chain being of equal length. The pulley is mounted on a frictionless horizontal spindle. Show that the position is unstable, but that it will become stable, for a small displacement, if a weight \(W\) is attached to the lowest point of the pulley, provided that \[ W > 2wa. \]
A string of length \(2l\) and of uniform density \(w\) is fixed at \(A, B\), two points distant \(2a\) at the same level. A weight \(W\) is hung on the string at the mid-point \(C\). Find the equations of the two portions of the string.
A machine gun of mass \(M\) contains shot of mass \(M'\) and stands on a horizontal plane. Shot is fired at the rate \(m\) per second with velocity \(u\) relative to the ground. If the coefficient of sliding friction between the gun and the plane is \(\mu\), show that the velocity of the gun backwards by the time the mass \(M'\) is fired is \[ \frac{M'}{M}u - \frac{(M+M')^2-M^2}{2mM}\mu g. \]
The end \(P\) of a straight rod \(PQ\) describes with uniform angular velocity a circle of centre \(O\), while the other end moves on a fixed line through \(O\) in the plane of the circle. The end \(Q'\) of an equal straight rod \(P'Q'\) moves on the same fixed line through \(O\). Prove that the velocities of \(Q, Q'\) are in the ratio \(QO:OQ'\).
A cyclist works at the constant rate of \(P\) horse-power. When there is no wind he can ride at 22 feet per second on level ground, and at 11 feet per second up a hill making an angle \(\sin^{-1}\frac{1}{5}\) with the horizon. The total mass of man and cycle is 180 lb. The resistance of the air is \(kv^2\) lb. weight, when the velocity of the man relative to the air is \(v\) feet per second; the other frictional forces are negligible. Find \(P\), and show that the speed of the cyclist when riding on level ground against a wind of 22 feet per second is between 10 and 10.5 feet per second.
Two equal smooth spheres moving along parallel lines in opposite directions with velocities \(u, v\) collide, the line of centres making an angle \(\alpha\) with the direction of motion. If, after the impact, their lines of motion are perpendicular, show that \[ \left(\frac{u-v}{u+v}\right)^2 = \sin^2\alpha+e^2\cos^2\alpha, \] where \(e\) is the coefficient of elasticity.
Find the equation whose roots are the squares of the differences of the roots taken in pairs of the cubic equation \(x^3+bx+c=0\). Hence or otherwise shew that if \(b\) and \(c\) are real the equation \(x^3+bx+c=0\) will have three real roots or only one real root according as \(4b^3+27c^2\) is negative or positive. Consider also the case in which \(4b^3+27c^2=0\).
(a) Find by the method of differences or otherwise the \(n\)th term and the sum to \(n\) terms of the series: \(1+4+11+26+57+120+\dots\). (b) Find the sum of the infinite series whose \(n\)th term is \(\displaystyle\frac{3n^2+3n+1}{n^3(n+1)^3}\).