A horizontal trough is formed by two planes both inclined at angles \(\theta\) to the horizontal. A uniform circular cylinder of weight \(W\) rests in contact with both planes at the bottom of the trough with its axis parallel to the line of intersection of the planes. Another circular cylinder of weight \(w\) rests in contact with this cylinder and one of the planes. The radius of the second cylinder is such that the axes are at the same horizontal level. All the surfaces are smooth. Show that equilibrium is only possible if \(w < W\), and find the reactions at the points of contact.
A man stands on an escalator which is descending at a steady speed \(u\), and initially he is at rest relative to the escalator. At the instant \(t=0\) he starts to walk up the escalator. His motion relative to the escalator is at first motion with uniform acceleration \(f\), but when he attains a speed \(2u\) relative to the escalator he continues to move relative to the escalator at this speed. Prove that at time \(t\) his displacement in space from his initial position is
\[ \frac{1}{2}f(t-t_0)^2 - \frac{1}{2}ft_0^2, \quad \text{for } 0
A train of mass 600 tons is originally at rest on a level track. It is acted on by a horizontal force \(X=\frac{1}{2}t\), where \(X\) is measured in tons weight and \(t\) in seconds. There is a resistance to motion of \(4\frac{1}{2}\) tons weight, and this resistance is independent of the speed of the train. Find the instant of starting \(t_0\), and prove that at the instant \(t=20\) the speed of the train is about 1.1 miles per hour. What is the horse-power required at \(t=20\)?
A particle is tied to a fixed point \(O\) by a light elastic string. The natural length of the string is \(a\), and the stretched length, when the particle hangs in equilibrium, is \(5a/4\). The particle is allowed to fall from rest at \(O\). Prove that the greatest depth below \(O\) reached by the particle is \(2a\). How long does the particle take to reach this depth after it is released from \(O\)?
A particle of mass \(m\) is projected vertically upwards in vacuo with speed \(u\). Prove that it returns to the point of projection after a time \(2u/g\). If the motion takes place in a resisting medium offering a resistance \(km|v|\) when the speed is \(v\), prove that the particle returns to the starting point after a time \(t_0\) given by the equation \[ 1-e^{-kt_0} = \frac{kgt_0}{g+ku}. \]
The end \(A\) of a light string \(AB\) is held fixed, and a particle of mass \(m\) is attached to the end \(B\). The particle moves in a horizontal circle with angular velocity \(\omega\). Prove that if the string is inelastic and of length \(l\), the inclination \(\alpha\) of the string to the downward vertical during the motion is given by the equation \[ \cos\alpha = \frac{g}{l\omega^2}. \] If the string is elastic, its natural length being \(a\), and if the tension required to double its length is \(kmg\), prove that the inclination is given by the equation \[ \cos\alpha = \frac{g}{a\omega^2} - \frac{1}{k}. \]
A thin flexible rope is wrapped \(n\) times round a rough post. Show that if the coefficient of friction between the rope and the post is everywhere equal to \(\mu\), a tension \(T\) at one end can balance a tension \(T e^{2\pi n \mu}\) at the other.
A cylinder of radius \(a\) is such that its centre of gravity \(G\) is at distance \(r\) from its axis. The cylinder can roll on a perfectly rough plane of inclination \(\alpha\) with its axis horizontal. Show that if \(r > a \sin\alpha\), equilibrium is possible in two different positions, and that in each of them the plane through \(G\) and the axis of the cylinder makes the same angle with the vertical. Show that one of these positions corresponds to stable equilibrium and one to unstable equilibrium.
Show that the principal axes of inertia at a corner of a uniform rectangular plate of sides \(2a, 2b\) make with the sides angles \(\theta\) and \(\frac{1}{2}\pi+\theta\) where \[ \tan 2\theta = 3ab/2(a^2-b^2). \]
Two equal spheres are at rest in a smooth tube bent in the form of a circle whose plane is horizontal. They are initially at opposite ends of a diameter when one of them is projected along the tube to collide with the other after time \(t\). If \(e\) is the coefficient of restitution, find the time that elapses between the first collision and the succeeding one.