A narrow straight tube of length \(2a\) has one end fixed and is made to rotate in a plane with constant angular velocity \(\omega\). A small bead is instantaneously at rest at \(t = 0\) at the mid-point of the tube, and the coefficient of friction in the tube is \(\frac{1}{3}\). If gravity can be neglected, show that the particle will reach the other end of the tube after time \((2/\omega) \log x\), where \(x\) is the larger positive root of the equation \(4x^2 - 10x^4 + 1 = 0\).
A particle of unit mass is describing an orbit, whose pedal equation is \(r = p/\sin^2 \phi\), under the influence of a central force \(F(r)\). Show that the value of the force at any point of the path is given by \[F = -\frac{1}{4} h^2 \frac{d}{dr}(r^{-2}),\] where \(r\) is the radius vector, \(p\) the perpendicular from the centre \(O\) onto the tangent, \(\phi\) is the speed of the particle, and \(h = pr\). The particle is projected from a point \(P\) under an attractive force \(2\mu r/r^3\), where \(\mu\) is a constant. Prove that if the velocity of projection has a certain value, to be found, the path will be a circle passing through the centre of force \(O\).
Two small spheres of masses \(m_1\) and \(m_2\) are in motion along the same straight line. Show that their kinetic energy may be written in the form \[\frac{1}{2}Mu^2 + \frac{1}{2}\mu v^2,\] where \(M = m_1 + m_2\), \(\mu^{-1} = m_1^{-1} + m_2^{-1}\), and \(u\) is the velocity of their centre of gravity and \(v\) is the velocity of one sphere relative to the other. If the spheres subsequently collide and their coefficient of restitution is \(e\), find the loss of kinetic energy.
The point of suspension of a simple pendulum \(AB\) of length \(l\) is \(A\), and the point \(A\) is caused to move along a horizontal straight line \(OX\) in such a way that \(OA = x(t)\) at time \(t\). If \(\theta\) is the inclination of the pendulum to the vertical, and \(g\) the acceleration of gravity, obtain the appropriate equation of motion. If \(\frac{d^2x}{dt^2}\) is constant and equal to \(f\), show that the pendulum can remain at a constant inclination \(\alpha\) to the vertical given by \(\tan \alpha = f/g\), and find the period of small oscillations about this position.
A light string \(ACE\), whose mid-point is \(C\), passes through two small smooth rings \(B\) and \(D\) at the same level and distance \(2a\) apart. At the points \(A\), \(C\), \(E\) of the string are attached masses each equal to \(m\). Initially \(C\) is at rest at \(O\) (the middle point of \(BD\), \(B\) and \(D\) are hanging vertically, and the system is set free. If the total length of the string is \(4l\), show that \(C\) will come to rest when it has fallen a distance \(4a/3\). Find also the speed of \(C\) when it has fallen a distance \(3a/4\) below \(O\).
A uniform heavy rod \(AB\) of length \(2a\) is suspended in equilibrium by two light strings \(OA\), \(OB\) each of length \(2a\). If the string \(OB\) is suddenly cut, find the initial angular acceleration of the rod and the new initial tension in the string \(OA\).
The tangents at the points \(B\), \(C\) on a conic are \(e\), \(f\) respectively; \(x\), \(y\) are the tangents from a point \(A\). Denoting the meet of \(e\), \(f\) by \((gf)\), and so on, prove that
A variable conic through fixed points \(K\), \(L\), \(M\), \(N\) meets a fixed line through \(N\) in \(P\). Prove that the envelope of the tangent at \(P\) is a conic inscribed in the triangle \(KLM\). Interpret this result when \(K\) and \(L\) are the circular points at infinity.
If \(n\) is a positive integer, show that $$\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^{n+2} & b^{n+2} & c^{n+2} \end{vmatrix}$$ has the value \((b-c)(c-a)(a-b)S\), where $$S = \sum a^r b^s c^t$$ summed over all values \(r\), \(s\), \(t\) satisfying \(r+s+t=n\). Prove a similar result for $$\begin{vmatrix} 1 & 1 & 1 & 1 \\ a & b & c & d \\ a^2 & b^2 & c^2 & d^2 \\ a^{n+3} & b^{n+3} & c^{n+3} & d^{n+3} \end{vmatrix}$$ and generalise the result.
The lines \(AB\) and \(A'B'\) are equal in length and lie in a plane. Show that \(A'B'\) can always be brought into coincidence with \(AB\) by either a rotation about a point in the plane or a translation. Show also that \(A'B'\) can be brought into coincidence with \(AB\) by a reflection in a suitably chosen line followed by translation parallel to the line. Prove that successive reflections of a plane figure in two non-parallel lines in the plane are equivalent to a rotation and that an odd number of reflections is equivalent to a single reflection followed by a translation.