One end of a uniform plank of weight \(W_1\) is smoothly hinged to one end of a uniform plank of weight \(W_2\). The structure stands upright with the free ends of both planks resting on a rough horizontal plane, the coefficient of friction being \(\mu\). The plank of weight \(W_1\) is inclined to the vertical at an angle \(\theta_1\) and the plank of weight \(W_2\) is inclined to the vertical at an angle \(\theta_2\). Calculate the reaction at the hinge. If the planks are on the point of slipping at both feet simultaneously, show that \[ W_1 \tan \theta_2 = W_2 \tan \theta_1, \] and also that \[ \mu = \sin \theta_1 \sin \theta_2 / \sin(\theta_1+\theta_2). \]
A uniform beam of length \(4l\) and weight \(4wl\) rests symmetrically on two supports at a distance \(2l\) apart, the points of support being in a horizontal line. A weight \(4wl\) is placed on the beam at a point between the supports, and at a distance \(l/2\) from one of them. Draw graphs showing the variation along the beam of the shearing force and of the bending moment.
A uniform rod \(AB\) of length \(d\) and weight \(W\) is smoothly pivoted at \(B\) to a fixed support and \(A\) is attached to one end of a light elastic string of unstretched length \(l\), the other end of which is fastened to a fixed peg \(C\) vertically above \(B\), where \(BC=h\) and \(h>d+l\). When the length \(x\) of the string exceeds \(l\), the tension in the string is \(\lambda(x-l)/l\). Show that the position of equilibrium in which \(AB\) is vertical, with \(A\) above \(B\), is stable if \[ 2\lambda h (h-d-l) > Wl(h-d). \] If a position of equilibrium exists in which \(AB\) is inclined to the vertical, show that in this position the length of the string is \(2\lambda hl / (2\lambda h - Wl)\).
A flexible chain hangs freely under gravity with its ends supported and is such that the tension \(T\) at any point \(P\) is given by \(T=kw\), where \(w\) is the weight per unit length at \(P\) and \(k\) is a constant. Show that \[ T=T_0 e^{y/k}, \] and that the length \(s\) of the arc measured from the lowest point \(O\) to \(P\) is given by \[ s = k \log \tan \left( \frac{x}{2k} + \frac{\pi}{4} \right), \] where \(x, y\) are the coordinates of \(P\) referred to horizontal and upward vertical axes through \(O\) and \(T_0\) is the tension at \(O\).
(i) How many degrees of freedom has
A particle of unit mass is projected vertically upwards from \(O\) with speed \(V\). The air resistance is given by multiplying the square of the speed of the particle by a constant \(k\). Show that the greatest height above \(O\) reached by the particle is \[ \frac{1}{2k} \log \left( 1 + \frac{kV^2}{g} \right). \] With what speed does the particle return to \(O\)?
A smooth straight tube is closed at one end \(O\), and is made to rotate about \(O\) in a vertical plane with constant angular velocity \(\omega\). A particle moves inside this tube. Initially the tube is vertically downwards and the particle is released from \(O\). If the particle is at distance \(r\) from \(O\) after a time \(t\), show that \[ \frac{d^2r}{dt^2} - \omega^2 r = g \cos \omega t, \] and hence that \[ 2\omega^2 r = g(\cosh \omega t - \cos \omega t). \]
A bead threaded on a fixed circular loop of wire lying in a vertical plane is set in motion from the lowest point of the wire with velocity \(V\) and first comes to rest at one end of the horizontal diameter of the wire. The radius of the loop is \(a\) and the coefficient of friction between the bead and the wire is \(\frac{1}{2}\). If \(v\) is the velocity of the bead when the line joining it to the centre of the loop makes an angle \(\theta\) with the downward vertical, show that \[ \frac{d}{d\theta} (v^2 e^\theta) = -ga e^\theta (2 \sin \theta + \cos \theta), \] and hence that \[ 2V^2 = ag(1+3\pi/2). \]
A particle is projected horizontally with velocity \(u\) from the lowest point of a fixed smooth hollow sphere of internal radius \(a\). Show that, if \(2ag < u^2 < 5ag\), the particle will leave the sphere when it is at a height \((u^2+ag)/3g\) above the lowest point. Show further that if \(2u^2=7ag\) the particle will subsequently strike the sphere at the lowest point.
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 \[ \frac{1}{4}mg(1-3\cos\theta)^2 + \frac{1}{4}ml\omega^2\cos\theta. \]