Problems

Prove Varignon's theorem, that the sum of the moments of two coplanar forces about any point in their plane is equal to the moment of their resultant about that point. Investigate whether the converse holds, namely, that if Varignon's theorem is true, then the resultant of two coplanar forces must be given by the parallelogram construction. A force in a given plane has moments \(G_1, G_2, G_3\), respectively, about points whose coordinates referred to axes in the plane are \((a_1, b_1), (a_2, b_2)\) and \((a_3, b_3)\). Determine the components \((X,Y)\) of the force referred to these axes and show that the force meets the \(x\)-axis in the point whose abscissa is \[ \begin{vmatrix} b_1 & a_1 & G_1 \\ b_2 & a_2 & G_2 \\ b_3 & a_3 & G_3 \end{vmatrix} \div \begin{vmatrix} b_1 & 1 & G_1 \\ b_2 & 1 & G_2 \\ b_3 & 1 & G_3 \end{vmatrix}. \]

The barrel of a gun of mass \(M\) is horizontal and of length \(l\); whilst a shell of mass \(m\) is being discharged from the gun the propelling gases exert a constant force \(P\) on the shell, and this force ceases as soon as the shell leaves the gun. From the instant of firing until the gun is brought to rest, recoil is resisted by a constant damping force \(R(

Show that the four points \((ct_i, c/t_i)\) \((i=1, 2, 3, 4)\) of the rectangular hyperbola \(xy=c^2\) are concyclic if and only if \(t_1 t_2 t_3 t_4 = 1\). \(A, B, C, D\) are the four points of intersection of a circle and a rectangular hyperbola. Prove that the six perpendiculars from the mid-points of the sides of the quadrangle \(ABCD\) to the opposite sides are concurrent.

If \[y = \frac{\log \{x + \sqrt{(1+x^2)}\}}{\sqrt{(1+x^2)}},\] verify that \[(1+x^2)\frac{dy}{dx} + xy = 1.\] Assuming that \(y\) can be expanded in a series of ascending powers of \(x\), prove that the series is \[x - \frac{2}{3}x^3 + \frac{2.4}{3.5}x^5 - \dots + (-1)^n \frac{2.4\dots2n}{3.5\dots(2n+1)}x^{2n+1} + \dots.\]

For each of the following, write down an equation of motion, giving your reasons fully, and deduce an expression for the velocity \(v\) after a time \(t\):

1. a drop of water, initially at rest and of mass \(M\), falling under gravity through a vacuum and losing mass by evaporation at a constant rate \(m\);
2. a drop of water, initially at rest and of mass \(M\), falling under gravity through a stationary cloud, and gaining mass at a constant rate \(m\) by coalescence with particles of the cloud. The frictional resistance to the motion is negligible.

The ends of a light spring of natural length \(2a\) and modulus \(\lambda\) are fixed at points \(A, B\) on a smooth horizontal table at distance \(4a\) apart, and a particle of mass \(m\) is fixed to the mid-point of the spring. Write down the equation of motion of the particle along the horizontal perpendicular bisector of \(AB\), and, by integrating this equation, prove that energy is conserved in the motion. Prove that, for oscillations of small amplitude, the period is approximately \(2\pi \sqrt{\left(\dfrac{ma}{\lambda}\right)}\).

\(P, Q, R\) are points on the sides \(BC, CA, AB\) of a triangle \(ABC\), and are not collinear. \(QR\) meets \(BC\) in \(L\), \(RP\) meets \(CA\) in \(M\), \(PQ\) meets \(AB\) in \(N\). Show that \(L, M, N\) are collinear if and only if \(AP, BQ, CR\) are concurrent. If \(AP, BQ, CR\) meet in \(O\), the line \(LMN\) may be called the polar of \(O\) with respect to the triangle \(ABC\). Show that in this case \(LMN\) is also the polar of \(O\) with respect to the triangle \(PQR\). Finally, if \(l, m, n\) are the polars of \(A\) with respect to \(OBC\), of \(B\) with respect to \(OCA\) and of \(C\) with respect to \(OAB\), show that \(LMN\) is the polar of \(O\) with respect to the triangle whose sides are \(l, m, n\).

Prove that, if \(x=r\cos\theta, y=r\sin\theta\) and \(\phi\) is any function of \(r\) and \(\theta\), \[\frac{\partial\phi}{\partial x} = \cos\theta \frac{\partial\phi}{\partial r} - \frac{\sin\theta}{r}\frac{\partial\phi}{\partial\theta},\] and obtain a corresponding expression for \(\partial\phi/\partial y\). Prove that, if \(\phi = r^{-n}\sin n\theta\), then \[\frac{\partial^2\phi}{\partial x^2} + \frac{\partial^2\phi}{\partial y^2} = 0.\]

A number of equal masses \(m\) are joined by light strings of length \(s\) so that the masses are at the angular points of a regular polygon, of side \(s\), inscribable in a circle of radius \(a\). The polygon lies on a smooth horizontal table and rotates steadily in its own plane with angular velocity \(\omega\) about its centre. Show that the tensions in the strings are \(ma^2\omega^2/s\) in absolute units. By considering the limiting case in which the number of sides tends to infinity, find the tension in a uniform string, of mass \(2\pi a \lambda\), which is in the form of a horizontal circle of radius \(a\) and is rotating in its own plane about its centre with uniform angular velocity \(\omega\). Verify this latter result by introducing centrifugal forces and applying the principle of virtual work to the corresponding statical problem.

Two gear wheels \(A\) and \(B\), of radii \(a, b\) and moments of inertia \(I, I'\) respectively, are mounted so as to be able to rotate without appreciable friction about their respective axes. The wheels are toothed and run permanently in mesh. A constant torque \(G\) is applied to \(A\) about its axis. Find

1. the tangential force between the wheels,
2. the angular acceleration of \(B\),
3. the number of revolutions made by \(B\) in acquiring from rest an angular velocity of \(N\) revolutions per second.