A uniform rod of mass \(m\) and length \(l\) is oscillating under gravity in a vertical plane, one end of the rod being attached to a fixed smooth hinge. The maximum angular velocity of the rod is \(\omega\). Show that, when the rod makes an angle \(\theta\) with the downward vertical, the vertical component of force exerted on the rod by the hinge is \[ \frac{1}{4}mg(1-3\cos\theta)^2 + \frac{1}{4}ml\omega^2\cos\theta. \]
Three forces of magnitudes \(la, mb\) and \(nc\) act at a point and are parallel to the sides (of lengths \(a, b\) and \(c\)) of a triangle \(ABC\). Find the magnitude of their resultant, and hence, or otherwise, show that the forces are in equilibrium if, and only if, \(l=m=n\) and the forces act in the directions of the sides taken in order.
A uniform rod of length \(2l\) and weight \(w\) per unit length rests on two supports on the same level, each support being at a distance \(a\) from the centre of the rod. Find expressions for the bending moment \(M\) at any point of the rod, and determine \(M^*\), the numerically greatest value of \(M\). If \(a\) is varied show that \(M^*\) is least if \(a = 0 \cdot 59l\).
A framework \(ABCD\) of four uniform rods, smoothly jointed together at \(A, B, C, D\), hangs freely from \(A\) and is kept in the form of a square by a light string joining \(A\) and \(C\). Each rod is of weight \(w\) and length \(a\), and a weight \(W\) is suspended from \(B\) and \(D\) by two light strings each of length \(\sqrt{2}a\). Find the tension in the string \(AC\).
An elastic string, which when unstretched is uniform, of length \(l\) and of weight \(w\) per unit length, hangs in equilibrium under gravity from one end \(O\) and supports a weight \(W\) at its other end. The tension in an element of the string of which the unstretched and stretched lengths are \(dx\) and \(dy\) is \(\lambda\left(\frac{dy}{dx}-1\right)\), where \(\lambda\) is constant. If the particle of the string which is at a distance \(x\) from the end \(O\) when the string is straight and unstretched is at a distance \(y\) from \(O\) when the string is suspended, find \(y\) as a function of \(x\), and show that the stretched length of the string exceeds \(l\) by \[ \frac{l}{\lambda}(W+\frac{1}{2}wl). \]
A smooth bead of mass \(m\) is free to slide on a circular wire of radius \(a\), which is fixed in a vertical plane. The bead is joined to the highest point of the circle by a light elastic string of modulus \(\lambda\) and natural length \(ka\) (where \(k<2\)). Show that there are three positions or one position of equilibrium with the string taut, according as \[ k < \frac{2\lambda}{\lambda+2mg} \quad \text{or} \quad k \ge \frac{2\lambda}{\lambda+2mg}. \] Investigate the stability of equilibrium in each case.
A bead \(P\) of unit mass moves without friction along a rigid straight wire. \(A\) is a point at a perpendicular distance \(a\) from the wire, \(r\) the distance \(AP\). The bead experiences a force of attraction towards \(A\) of magnitude \(k^2r^{-3}\), where \(k\) is a constant. At the time \(t=0\) the bead is at \(r=a\) and has the speed \(k/a\). Show that it reaches its greatest deceleration at the time \[ \frac{a^2}{k}\left(\frac{1}{4}\pi + \frac{1}{2}\log 3\right). \]
Two equal masses \(m\) move in straight lines against a resistive force \(kv\), where \(v\) is the speed and \(k\) a constant. The first mass is propelled by a constant force, the second by an engine doing work at a constant rate. The limiting speed \(w\) is the same for both masses. Show that the times required for the two masses to attain a speed \(v\) starting from rest differ by the amount \[ t_1 - t_2 = \frac{m}{2k}\log\frac{w+v}{w-v}, \] and that the respective distances travelled in these times differ by \[ x_2 - x_1 = \frac{mw}{2k}\log\left(1-\frac{v^2}{w^2}\right). \]
Particles are projected under gravity in a vertical plane from a point \(O\) on level ground with initial velocity \(v\) at all angles of elevation. Show that the region above the ground covered by all the trajectories is bounded by the trajectory of a particle which, at a certain point vertically above \(O\), moves horizontally with velocity \(v\). \newline Find also the region covered by the ascending parts of all trajectories and the region covered by all trajectories with angle of elevation less than 45\(^\circ\). Indicate the position of the various regions on a diagram.
Express in polar co-ordinates \(r, \theta\) the radial and transverse components of velocity and acceleration of a particle moving in a plane. \newline A particle of mass \(m\) moves in a plane under the action of a force of magnitude \(m\lambda v/r\), where \(v\) is its speed, \(r\) its distance from a fixed point \(O\) and \(\lambda\) a constant. The force acts at right angles to the direction of motion and deflects the particle to the left. Initially the particle is at a point \(P\) at distance \(a\) from \(O\) and its component of velocity perpendicular to \(OP\) is \(\lambda\), the motion being anticlockwise around \(O\). Show that the radial and transverse components of the velocity remain constant and find the polar equation to the path.