A uniform solid sphere, which is initially rotating with angular velocity \(\omega\) about a horizontal diameter, is gently placed on a rough horizontal plane. If \(\mu\) is the coefficient of friction, show that the sphere moves a distance \(2\omega^2 a^2/49\mu g\) before slipping ceases, and that the angular velocity is then \(2\omega/7\).
Prove that, if all the numbers involved are real, the function \(f(x)\) defined by \[ f(x) = \frac{ax^2+2bx+c}{x^2+k} \] is capable of all real values if \[ a^2k^2+2k(2b^2-ac)+c^2 < 0. \] Prove that this inequality implies the two \[ b^2>ac, \quad k<0, \] and investigate the conditions for the existence of two limiting values between which
Complex numbers \(z_r (z_r = x_r+iy_r)\) are represented in the Argand diagram by points \(P_r\) with co-ordinates \((x_r, y_r)\). Prove that
A match between two players \(A\) and \(B\) is won by whoever first wins \(n\) games. \(A\)'s chances of winning, drawing or losing any particular game are \(p, q\) and \(r\) respectively. Prove that his chance of winning the match is \(p^2(p+3r)/(p+r)^3\) if \(n\) is 2, and \[ p^3(p^2+5pr+10r^2)/(p+r)^5 \] if \(n\) is 3.
The lines joining a point \(P\) to the vertices of a triangle \(ABC\) meet the opposite sides in \(L, M, N\). A variable conic through \(L, M, N, P\) meets \(BC\) again in \(U\), and the tangent to the conic at \(P\) meets \(BC\) in \(V\). Prove that \(U, V\) are harmonic conjugates with respect to \(B, C\). \newline A line \(p\) through \(P\) meets \(BC, CA, AB\) in \(A', B', C'\), and \(L'\) is the harmonic conjugate of \(A'\) with respect to \(B\) and \(C\); \(M', N'\) are similarly defined. Prove that \(L, M, N, L', M', N'\) lie on a conic which touches \(p\) at \(P\).
Prove that, if \(f(x)\) is a function of \(x\) which has a derivative \(f'(x)\) for all values of \(x\) between \(a\) and \(b\) inclusive, and if \(f(a)=f(b)\), there is at least one value \(\xi\) between \(a\) and \(b\) for which \(f'(\xi)=0\). \newline Deduce from this theorem that, for some \(\xi\) between \(a\) and \(b\), \[ \text{(i)} \quad \frac{\phi(b)-\phi(a)}{\psi(b)-\psi(a)} = \frac{\phi'(\xi)}{\psi'(\xi)}, \] and that, for another \(\xi\), \[ \text{(ii)} \quad \frac{\phi(\xi)-\phi(a)}{\psi(b)-\psi(\xi)} = \frac{\phi'(\xi)}{\psi'(\xi)}, \] where in each case it is assumed that \(\phi'(x), \psi'(x)\) exist for all values of \(x\) between \(a\) and \(b\) inclusive, and that \(\psi'(x)\) does not vanish for any \(x\) between \(a\) and \(b\).
The framework \(ABCDEFGH\) consists of eight equal uniform heavy rods smoothly jointed at their ends, and is maintained in the shape of a regular octagon by light struts \(AF, AG, BD\) and \(BE\). The framework is in equilibrium, suspended from the mid-point of \(AB\). State which (if any) of the stresses in the struts are thrusts and which tensions, and determine the ratio of the magnitudes of the stresses in \(AF\) and \(AG\).
A uniform chain of weight \(w\) per unit length hangs in equilibrium under gravity over a rough circular cylinder of radius \(a\). The chain, the length of which exceeds \(\pi a\), lies in a plane perpendicular to the horizontal axis of the cylinder with one end \(A\) on the same level as the axis, and \(T\) is the tension in the chain at the point at an angular distance \(\theta\) (less than \(\pi\)) from \(A\). If the chain is on the point of slipping in the direction in which \(\theta\) increases, prove that \[ \frac{d}{d\theta}(Te^{-\mu\theta}) = wa(\cos\theta+\mu\sin\theta)e^{-\mu\theta}, \] where \(\mu\) is the coefficient of friction, and hence find the length of the part of the chain not in contact with the cylinder.
A solid body of uniform density consists of a circular cone of perpendicular height \(4a\), to whose base (which is a circle of radius \(a\)) is attached a hemisphere of radius \(a\), the plane surface of the hemisphere coinciding with the base of the cone. The body is free to rotate about a fixed horizontal axis through the vertex of the cone. Find the length of the simple pendulum having the same period for small oscillations.
Two equal balls \(A, B\) are placed on the baulk line \(PQ\) of a billiard table, which may be regarded as a perfectly rough horizontal plane \(JKLM\) of rectangular shape. \(PQ\) is parallel to the side \(JK\) and is distant \(5l\) from it, \(P\) being a point of the side \(JM\). \(A\) is equidistant from \(P\) and from \(B\), and \(PB=2l\). The balls are struck simultaneously, \(A\) rolling with velocity \(V\) along a line parallel to \(PJ\) and \(B\) rolling so as to strike \(A\), the line of centres being at the moment of impact in the direction of the motion of \(B\). After the collision, \(A\) rolls on into the pocket at \(J\). Find the initial velocity of \(B\) in magnitude and direction, assuming that the coefficient of restitution between the balls is \(\frac{1}{2}\), and that their diameters can be neglected.