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 \(M/(m+M)\).
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?
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.
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.
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 \(t\).
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 \(r \to \infty\).
If \(B(p,q) = \int_0^1 x^{p-1} (1-x)^{q-1} dx\) for \(p>0, q>0\), prove that \begin{align*} B(p,q) &= B(p+1,q) + B(p,q+1), \\ B(p,q) &= B(q,p). \end{align*} Prove that, if \(p\) and \(q\) are integers, \[ B(p,q) = \frac{(p-1)!(q-1)!}{(p+q-1)!}. \]
\(ABC\) is a given triangle and \(P\) is a general point in its plane. The lines \(PA, PB, PC\) meet \(BC, CA, AB\) in \(X, Y, Z\), respectively, and \(YZ, ZX, XY\) meet \(BC, CA, AB\) in \(L, M, N\), respectively. Show that \(L, M, N\) lie on a line \(p\). Show also that, in general, a given line \(p\) may be derived in this way from one and only one point \(P\). Calling \(P\) and \(p\) pole and polar with respect to the triangle \(ABC\), prove that, if a point \(P\) describes a straight line \(q\), its polar \(p\) envelops in general a conic \(\Sigma\). Show also that \(q\) has the same pole with respect to \(ABC\) and with respect to \(\Sigma\).
A uniform circular disc of radius \(a\) and mass \(M\) can turn in its own plane about a fixed horizontal axis through the centre. A light inextensible string lies in a rough groove in the edge of the disc; the end portions of the string are vertical and masses \(M, 2M\) are carried at the ends. If these masses move vertically, show that the angular acceleration of the disc is \(2g/7a\). Find also the tensions in the vertical portions of the string. Find the magnitude of the frictional couple that must be applied to the disc in order that its angular acceleration may be \(g/7a\).
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.