A bead of mass \(m_1\), a light spiral spring, and a bead of mass \(m_2\) are threaded in that order on a smooth, straight horizontal wire. If initially the bead \(m_1\) is moving towards the spring with speed \(U\), and the bead \(m_2\) is at rest, show that ultimately the bead \(m_2\) acquires a speed \(2m_1 U / (m_1+m_2)\).
Shew that the shearing stress in a rod (not necessarily of negligible weight) is continuous except at a concentrated load, and that the bending moment is continuous at such a point. A wire weighing \(w\) per unit length is in the form of a semicircle of radius \(a\). It rests, in a vertical plane, with its ends on a smooth horizontal table. Show that at a point whose angular distance from the highest point is \(\beta\) the bending moment is \[ \frac{wa^2}{2}(\pi - \beta\sin\beta - \cos\beta). \] Shew also that the maximum value of the shearing stress occurs where \(\beta \tan\beta = 1\). [The centre of mass of a circular arc of angle \(2\alpha\) is distant \((a\sin\alpha)/\alpha\) from the centre.]
If \[ I_n = \int_0^{\frac{1}{2}\pi} (a^2 \cos^2\theta + b^2 \sin^2\theta)^n d\theta, \] where \(a\) and \(b\) are positive and \(n\) is real, prove by a transformation of the type \(\tan\theta = \lambda \tan\phi\), or otherwise, that \[ I_n = (ab)^{2n+1} I_{-n-1}. \] Hence, or otherwise, evaluate \[ \int_0^{\frac{1}{2}\pi} \frac{\cos^2\theta}{(1+\sin^2\theta)^3} d\theta. \]
Sketch the locus of the point \(\left( \frac{t}{1+t^3}, \frac{t^2}{1+t^3} \right)\) as \(t\) varies. Find the condition for the points given by \(t=t_1, t_2, t_3\) to be collinear. If the six vertices of a complete quadrilateral lie on this curve, prove that the values of \(t\) corresponding to any two opposite vertices are equal in magnitude and of different signs. Prove also that the tangents at two such vertices meet on the curve.
A uniform rod \(AB\) of length \(2a\) is suspended from a fixed point \(O\) by two light elastic strings \(OA, OB\), each of natural length \(a/\sqrt{2}\). When the system is in equilibrium with the rod horizontal each of the strings is inclined at 45\(^\circ\) to the vertical. Prove that, if the rod makes small vertical oscillations, remaining horizontal, the period of oscillation is \(\frac{2\pi}{n}\) where \(n^2 = \frac{3g}{2a}\).
A particle of mass \(m\) is projected vertically upwards with velocity \(U\). If the air resistance is \(mk\) times the square of the speed, shew that, if \(g/k=V^2\), the particle will be instantaneously at rest after a time \((V/g) \tan^{-1}(U/V)\). Shew also that it will reach its starting point with speed \(VU/(U^2+V^2)^{\frac{1}{2}}\).
Prove that, if \(f(x,y)\) is a function of \(x^2+y^2\) only, it satisfies the identical relation \[ y \frac{\partial f}{\partial x} = x \frac{\partial f}{\partial y}. \] By changing to polar coordinates, or otherwise, prove conversely that, if \(f(x,y)\) satisfies this relation identically, it must be a function of \(x^2+y^2\) only.
A system of curves is given by the equation \(f(x,y,c) = 0\), where \(c\) is a variable parameter. Show that, in general, the curve obtained by eliminating \(c\) from \(f=0, \partial f/\partial c=0\) touches each curve of the system. \(OX, OY\) are axes of coordinates, not necessarily rectangular. From a point \(P\) on a plane curve lines parallel to \(OY, OX\) are drawn to cut \(OX, OY\) at \(M, N\). \(Q\) is the point of contact of \(MN\) with its envelope, and the tangent at \(P\) to its locus meets \(MN\) at \(R\). Prove that \(OQ, OR\) are harmonic conjugates with respect to \(OX, OY\).
The coefficient of viscosity \(\eta\) of a fluid has dimensions --1 in length, 1 in mass and --1 in time. If for a certain fluid \(\eta\) has the value 0.23 in the centimetre-gram-second system, find its value in the metre-kilogram-minute system. If a sphere of radius \(a\) and weight \(W\) falls through a fluid whose coefficient of viscosity is \(\eta\), its speed approaches a terminal speed \(v\). Assuming that no other variables are involved, find by a consideration of dimensions how \(v\) depends on \(a, W\) and \(\eta\).
A rigid lamina of mass \(M\) moves in its own plane. Shew that the kinetic energy is the same as that of a particle of mass \(M\) having the same motion as the centre of mass \(G\), together with the kinetic energy that the lamina would have if \(G\) were fixed and the lamina rotated about \(G\) with the same angular velocity. A hollow ring is made of uniform thin sheet metal, and its form may be regarded as generated by the revolution of a circle of radius \(a\) about a line lying in the plane of the circle at a distance \(l\) (\(>a\)) from its centre. The ring hangs symmetrically on a fixed horizontal knife-edge and makes small oscillations in such a way that the mass centre moves at right angles to the knife-edge. There is no slipping at the knife-edge. Shew that the period is the same as that of a simple pendulum of length \[ (4l^2 - 4al + 5a^2)/2(l-a). \]