Problems

A homogeneous sphere is set rotating about a horizontal axis. It is projected in the direction of this axis on a horizontal table. The coefficient of friction between the sphere and the table is \(\mu\). Discuss the subsequent motion.

Explain the method of inversion in electrostatic problems. Find an expression for the potential round an insulated charged conductor which consists of the larger parts of two equal spheres which cut one another in a re-entrant angle of 60 degrees.

A soap film is attached to fixed wires in the form of one or more closed curves. Assuming that the film takes such a form as to render its area a minimum consistent with the given boundary conditions prove, by considering the variation of the integral \[ \iint \sqrt{1 + \left(\frac{\partial z}{\partial x}\right)^2 + \left(\frac{\partial z}{\partial y}\right)^2} dx dy, \] that the principal radii of curvature at any point of the film are equal in magnitude but opposite in sign. If the bounding wires are parallel circles whose planes are perpendicular to the line joining their centres prove that the meridian curve of the film is a catenary.

Find a differential equation which represents the path of a ray through a medium whose refractive index, \(\mu\), is a function of \(r\), the distance from a fixed centre. In the case when \(\mu = Cr^{-m}\), and \(C\) is a constant, show that the deviation of any ray in a portion of the path which subtends an angle \(\theta\) at the centre, is \(m\theta\).

An infinite circular cylinder of radius \(b\) and uniform density \(\sigma\) is surrounded by fluid of density \(\rho\). The outer boundary of the fluid is a concentric circular cylinder of radius \(a\). The outer cylinder is caused to execute small oscillations of amplitude \(\alpha\) in a direction perpendicular to its length. Show that the amplitude \(\beta\) of the oscillations of the inner cylinder is \[ \beta = \frac{2a^2\rho}{a^2(\sigma+\rho)+b^2(\sigma-\rho)}. \]

Any two perpendicular diameters \(POP'\), \(QOQ'\) of an ellipse are drawn; shew that the four lines \(PQ, PQ', P'Q, P'Q'\) touch a fixed circle with centre \(O\). Generalise the theorem by projection, and deduce (or prove otherwise) the following theorem: Any two chords \(POP'\), \(QOQ'\) of a conic are drawn through a fixed point \(O\), so as to be harmonically conjugate with respect to two fixed lines \(OA, OB\). Prove that the lines \(PQ, PQ', P'Q, P'Q'\) touch a fixed conic, which touches \(OA, OB\) at the points \(A, B\) where they meet the polar of \(O\) with respect to the given conic.

Shew that if two points on a bar are constrained to move along two perpendicular straight lines, the locus of any other marked point on the bar is an ellipse of which the two given lines are the principal axes.

1. [(i)] Generalise the theorem, making two angles of a triangle move on any two fixed straight lines, and proving that the locus of the third angle is an ellipse.
2. [(ii)] A heavy bar \(PQ\) is constrained to rest with its upper end \(P\) against a smooth vertical wall \(BB'\) and its lower end on the lower half of a smooth ellipse of which \(BB'\) is the minor axis, and the axes are equal to \(2PQ, 2GQ\), where \(G\) is the centre of gravity of the bar. Shew (from the principle of energy or otherwise) that the bar can rest in equilibrium at any inclination to the horizontal.

If \(\lambda_1, \lambda_2\) are the roots of the equation in \(\lambda\), \[ \begin{vmatrix} a-\lambda, & b \\ c, & d-\lambda \end{vmatrix} = 0, \] verify that \(\alpha_1=b/(\lambda_1-a), \alpha_2=b/(\lambda_2-a)\) are the roots of the equation in \(x\), \[ cx^2+(d-a)x-b = 0. \] Shew that the equation \(y = (ax+b)/(cx+d)\) can be written in the form \[ \frac{y-\alpha_1}{y-\alpha_2} = \frac{\lambda_2}{\lambda_1} \left(\frac{x-\alpha_1}{x-\alpha_2}\right), \] except when \(\lambda_2 = \lambda_1\); and that in this exceptional case \[ \frac{1}{y-\alpha_1} = \frac{1}{x-\alpha_1} + \frac{c}{\lambda_1}. \]

Give a discussion of the method of ``proportional parts'' as applied to interpolation in mathematical tables; and by considering the function \[ F(x) = f(x) - f(a) - \frac{x-a}{b-a}\{f(b)-f(a)\} - C(x-a)(x-b), \text{ or otherwise,} \] shew that the error in the value of \(f(c)\) as calculated from the tabular values given for \(x=a, x=b\), is equal to \[ \frac{1}{2}(b-c)(c-a)f''(\gamma) \] in excess of the true value, where \(c\) and \(\gamma\) lie between \(a\) and \(b\). Hence or otherwise determine whether the method can be applied safely to the four figure tables supplied, in the following cases:

1. [(i)] to find \(\log 1.45\) from \(\log 1.40\) and \(\log 1.50\),
2. [(ii)] to find \(\cot 1^\circ 45'\) from \(\cot 1^\circ 40'\) and \(\cot 1^\circ 50'\).

Starting from the equations \[ dx = \rho d\phi \cos\phi, \quad dy = \rho d\phi \sin\phi, \] shew that the expansions of the coordinates of a point \(P\) on a curve in powers of \(\phi\), the inclination of the tangent at \(P\) to the tangent at \(O\), are given by \[ x = \rho_0 \phi - \frac{1}{2}\rho_1\phi^2 + \frac{1}{6}(\rho_2 - \rho_0)\phi^3, \quad y = \frac{1}{2}\rho_0\phi^2 + \frac{1}{3}\rho_1\phi^3, \] where terms containing \(\phi^4\) are neglected, and \(\rho_0, \rho_1, \rho_2\) are the values of \(\rho, d\rho/d\phi\) and \(d^2\rho/d\phi^2\) at \(O\). Expand \(x \cot\phi + y\) in powers of \(\phi\) as far as the term in \(\phi^2\). The normals at \(O\) and \(P\) meet in \(N\); shew that in general \((ON - \rho_0)\) is equal to \(\frac{1}{2}(\rho - \rho_0)\) and is of order \(\phi\); but that when \(\rho_1=0\), the value is \(\frac{1}{4}(\rho - \rho_0)\), and is of order \(\phi^2\).