Problems

Four variables \(u, t, p, v\) are such that any one of them can be expressed as a function of any two of the others. Prove that \[ \left(\frac{\partial u}{\partial t}\right)_p = \left(\frac{\partial u}{\partial t}\right)_v + \left(\frac{\partial u}{\partial v}\right)_t \left(\frac{\partial v}{\partial t}\right)_p, \] where \(\left(\frac{\partial u}{\partial t}\right)_v\), for example, means that \(u\) is expressed as a function of \(t\) and \(v\), and \(v\) is kept constant in the differentiation.

A uniform beam of length \(2a\) and weight \(w\) per unit length rests symmetrically and horizontally on two supports at a distance \(2(a-b)\) apart, a length \(b\) of the beam projecting beyond each of the supports. Calculate the bending moment \(M\) at any point of the beam, illustrating your result graphically, and find where \(M\) is greatest, distinguishing between the two cases \(b \lesseqgtr (\sqrt{2}-1)a\). Make a diagram showing the dependence of the greatest value \(\bar{M}\) of \(M\) on \(b\). At what points must the beam be supported in order that \(\bar{M}\) shall be as small as possible?

Three light inextensible strings \(AB, BC, CA\) are respectively of lengths \(a, a, a\sqrt{2}\), and are knotted together at \(A, B, C\). Masses \(m, M\) are carried at \(B, C\) respectively, and are moving on a smooth horizontal table, so that \(ABC\) rotates freely about the knot \(A\), which is pinned to a point of the table. If the string \(AC\) is cut, show that the ratio of the tensions in \(BC\) and \(AB\) immediately afterwards is \(M/(m+M)\).

Prove that, in general, two conics of a given confocal system pass through an arbitrary point \(P\) of the plane, and that they cut orthogonally at \(P\). If the tangents at \(P\) to the two confocals through \(P\) meet one of the axes of the confocal system in \(X, X'\), and the other axis in \(Y, Y'\), prove that, for all positions of \(P\), the circle \(PXX'\) belongs to a certain coaxal system and the circle \(PYY'\) to the orthogonal coaxal system.

When \(x=a\), the functions \(f(x)\), \(g(x)\), \(f'(x)\) and \(g'(x)\) have the values \(0, 0, b\) and \(c\) respectively. Prove that, if \(c \ne 0\), \(f(x)/g(x)\) tends to \(b/c\) as \(x\) tends to \(a\). Evaluate the limits as \(x\) tends to \(0\) of \[ \text{(i) } \frac{3^x - 3^{-x}}{2^x - 2^{-x}}, \quad \text{(ii) } \frac{1}{x} \int_0^x \sqrt{(3t^2+4)} \, dt. \]

Two particles \(A, B\) are attracted to one another with a force of magnitude \(\lambda r^{-2}\), where \(\lambda\) is a constant and \(r\) is the distance between \(A\) and \(B\). Explain what is meant by the statement that the particles have a potential energy \(-\lambda r^{-1}\). Initially \(A\) is at rest and \(B\) is moving at a great distance from \(A\) along a straight line which passes close to \(A\). When the particles have approached one another and again separated to a great distance, the directions of motion of \(A, B\) make angles \(\alpha, \beta\) respectively with the initial direction of motion of \(B\). By considering the conservation of energy and momentum, prove that the masses of the particles are in the ratio \(\sin\beta : \sin(\beta-2\alpha)\). Show that the same result holds if the potential energy \(V(r)\) of the particles is any function of \(r\) such that \(V(r) \to 0\) as \(r \to \infty\).

Obtain the equation of motion of a simple pendulum of length \(l\), \[ l\frac{d^2\theta}{dt^2} + g \sin\theta = 0, \] and deduce that \[ \tfrac{1}{2}l \left(\frac{d\theta}{dt}\right)^2 - g \cos\theta = \text{constant}. \] If initially \(\theta=0\) and \(l\left(\frac{d\theta}{dt}\right)^2 = 4g\), prove that \(\sin\frac{1}{2}\theta = \tanh\sqrt{\frac{g}{l}}t\). Illustrate by a rough graph how \(\theta\) varies with \(t\).

If \(DD'\) is a diameter of a rectangular hyperbola and \(P\) any point on the curve, show that the normal at \(P\) meets the perpendicular bisectors of \(PD\) and \(PD'\) in points equidistant from \(P\). Hence, or otherwise, obtain a construction for the tangent at a given point of a rectangular hyperbola when one diameter is given.

Defining a cycloid as the locus of a point on the circumference of a circle which rolls along a straight line, prove that the centres of curvature at points of a cycloid lie on an equal cycloid.

A uniform chain of mass \(\rho\) per unit length and length \(2a\alpha\) can slide in a smooth tube bent into the form of a circle of radius \(a\) and fixed in a vertical plane. At the instant \(t\) the radii drawn to the ends of the chain make angles \(\beta+\alpha, \beta-\alpha\) with the downward vertical. Obtain the equation \[ a\alpha \frac{d^2\beta}{dt^2} + g \sin\alpha \sin\beta = 0. \] Find the tension in the chain at the end of the radius making an angle \(\beta+\theta\) with the downward vertical.