A particle is suspended from a fixed point by a light spring. If \(c\) is the extension of the spring when the particle hangs in equilibrium, and \(2\pi/\omega\) is the period of small vertical oscillations when equilibrium is disturbed, show that \(c\omega^2 = g\). From this particle is now suspended a second particle, of the same mass, by a similar spring. The particles are set in motion in a vertical line. Denoting the extensions of the upper and lower springs by \(2c + x\) and \(c + y\) respectively, write down the equations of motion. Show that two periodic motions each of the form $$x = a \cos \omega t, \quad y = b \cos \omega t$$ are possible, the frequencies being given by $$(\omega/\omega_0)^2 = \frac{1}{2}(3 \pm \sqrt{5}).$$ Find the corresponding values of \(b/a\).
A force \(\mathbf{F}\) acts at a point whose position vector from \(O\) is \(\mathbf{r}\). Define the moment of \(\mathbf{F}\) about \(O\) and the work done by \(\mathbf{F}\) in a displacement \(\delta \mathbf{r}\) of the point of application. A number of forces act at points of a rigid sheet of material, and are coplanar with the sheet. Deduce from your definitions the following. (If you express your definitions in terms of scalar or vector products, you should prove any properties of these products on which your deductions depend.) (i) If the sheet is given a uniform displacement \(\delta \mathbf{a}\) the total work done by the forces is equal to the work done by the resultant force \(\mathbf{F}\) in the displacement \(\delta \mathbf{a}\). (ii) If the sheet is given two uniform displacements \(\delta \mathbf{a}\) and \(\delta \mathbf{b}\) in succession, the total work done by the forces is equal to the work done by \(\mathbf{F}\) in the resultant displacement \(\delta \mathbf{a} + \delta \mathbf{b}\). (iii) If the sheet is given a small rotation \(\delta \theta\) about \(O\) the work done by the forces is \(L\delta \theta\), \(L\) being the total moment of the forces about \(O\). Find an expression for the work done by the forces in a small rotation \(\delta \theta\) about the point whose position vector is \(\mathbf{p}\).