Problems

A uniform heavy rigid beam \(AB\) of length \(2l\) and weight \(W\) rests in a horizontal position on two supports, one at the end \(B\) and the other at a point \(C\) between \(A\) and \(B\), while from the free end \(A\) hangs a weight \(W\). \newline For any given possible value \(c\) of \(AC\) obtain expressions for the bending moment and shearing force at any point \(X\) of the beam in terms of \(x\) the distance \(AX\). Illustrate your results by a sketch and indicate points of zero and maximum bending moment. \newline Prove that when \(C\) is chosen so that the beam is about to turn round the support at that point, the greatest value of the bending moment is \(\frac{9}{16}Wl\).

A system of three uniform rods \(AB, BC, CD\) of unequal lengths freely jointed at \(B\) and \(C\) is suspended freely from points \(A\) and \(D\) not necessarily on the same horizontal level. If the weights of \(AB\) and \(CD\) are equal, show that in equilibrium the mid-point of \(BC\) is at its lowest possible position.

A uniform flexible chain of length \(l\) and total weight \(wl\) has one end \(A\) attached to a fixed point. The other end \(B\) is held in position by the application of a horizontal force \(P\). \newline Prove that \(P=w \frac{(l^2-h^2)}{2h}\), and that the tension of the chain at \(A\) is \(w \frac{(l^2+h^2)}{2h}\), where \(h\) is the vertical distance \(A\) is above \(B\).

A weight is suspended from a fixed point \(O\) by a light flexible elastic string of natural length \(l\). When in equilibrium the weight is at a point \(A\) distant \(l+a\) below \(O\). Show that the period of vertical oscillations about \(A\) with semi-amplitude less than \(a\) is \(2\pi\sqrt{a/g}\). \newline [Hooke's law may be assumed to hold for any value of the extension.] \newline The weight is released from rest when at a point \(P\) below \(A\) so that it eventually rises to a point \(Q\) at a distance \(l-c\) (where \(c\) is positive) below \(O\) before beginning to fall. Show that the time taken to travel from \(P\) to \(Q\) is \[ \sqrt{\frac{a}{g}}\left\{\frac{\pi}{2} + \sin^{-1}\sqrt{\frac{c}{a+2c}} + \sqrt{\frac{a+2c}{a}}\sqrt{\frac{2c}{a}}\right\}. \]

A smooth thin horizontal straight rod rotates in a horizontal plane with constant angular velocity \(\omega\) about a fixed point \(O\) of itself. A smooth bead threaded on the rod is projected when at a distance \(a\) from \(O\) with a velocity \(u\) relative to the rod towards \(O\). Prove that, if \(u< a\omega\), the bead comes to momentary rest relative to the rod when at a distance \(\sqrt{a^2-u^2/\omega^2}\) from \(O\) and will again be at distance \(a\) from \(O\) after a time \(\frac{1}{\omega}\log_e\frac{a\omega+u}{a\omega-u}\). \newline If \(u> a\omega\), show that there is no position of relative rest and that the particle will again be at distance \(a\) from \(O\) after a time \(\frac{1}{\omega}\log_e\frac{u+a\omega}{u-a\omega}\). \newline What happens if \(u=a\omega\)?

A particle is released from rest at a point of a smooth thin tube in the form of a parabola held fixed in a vertical plane with axis vertical and vertex downwards, and moves under gravity in the tube. Prove that for the given motion the vertical component of the pressure of the particle on the tube is inversely proportional to the square of the height of the particle above the directrix of the parabola.

A particle moving under gravity in a medium offering resistance proportional to the fourth power of the velocity, if released from rest will attain a final velocity \(u\) vertically downwards, and if projected with the same velocity \(u\) vertically upwards will rise to a height \(h\) before beginning to fall. Prove that if released from rest and allowed to fall the particle will attain a velocity of \(\frac{1}{2}u\) when it has fallen a distance \(\frac{2}{5\pi}h\log_e 3\).

Two equal uniform smooth spheres can move on a smooth horizontal table without rolling and the coefficient of restitution for impact between them is \(e\). Show that the greatest deviation \(\delta\) that can be produced in the velocity of one sphere by collision with the other when it is stationary is given by \((3-e)\sin\delta=1+e\).

A pendulum consists of a rigid uniform wire of negligible thickness in the form of a circle of radius \(a\) and centre \(C\), and mass \(M\). Two particles of mass \(m\) are attached rigidly to the circumference of the circle at two points \(A\) and \(B\). If \(O\) is a point of the circumference on the diameter perpendicular to \(AB\), show that the pendulum swinging about \(O\) in the plane of the circle has a period for small oscillations which is independent of the length \(AB\) and equal to that of a simple pendulum of length \(2a\).

Solve the system of equations: \begin{align*} x+y+z &= a, \\ x+\omega y + \omega^2 z &= b, \\ x+\omega^2 y + \omega z &= c, \end{align*} where \(\omega\) is a complex cube root of unity, expressing \(x, y\) and \(z\) as simply as you can in terms of \(a, b, c\) and \(\omega\). \newline Assuming \(a, b\) and \(c\) to be real, show that \(x\) is real, and that \(y\) and \(z\) are conjugate complex numbers. Prove also that \[ x^2 + |y|^2 + |z|^2 = x^2+2yz = \frac{1}{3}(a^2+b^2+c^2), \] where \(|y|\) and \(|z|\) are the moduli of \(y\) and \(z\). Find the area of the triangle formed by the points representing the numbers \(x, y, z\) on the Argand diagram.