Problems

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

1. [(a)] a particle free to move along a given line,
2. [(b)] a lamina free to move in its own plane,
3. [(c)] a rigid body free to move in the most general manner?

(ii) A particle moves along a given straight line, and its distance at time \(t\) from a particular point of the line is \(x(t)\). Show that \[ \frac{d^2x}{dt^2} = - \left(\frac{dx}{dt}\right)^3 \frac{d^2t}{dx^2}. \] Hence, or otherwise, show that, if \(t\) is a quadratic function of \(x\), the acceleration of the particle is inversely proportional to the cube of its distance from a certain point of the line.

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 \[ \frac{1}{4}mg(1-3\cos\theta)^2 + \frac{1}{4}ml\omega^2\cos\theta. \]

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.