A cylinder of any oval cross section rests in equilibrium on a horizontal plane. Find the maximum height of the centroid in order that equilibrium may be stable. An elliptic cylinder rests in stable equilibrium on a horizontal plane. Prove that the equilibrium cannot be made unstable by placing a particle on the highest generator if the eccentricity of the cross section is \(> 1/\sqrt{2}\).
Explain clearly what is meant by relative velocity. The line joining two points \(A, B\) is of constant length \(a\) and the velocities of \(A, B\) are in directions which make angles \(\alpha\) and \(\beta\) respectively with \(AB\). Prove that the angular velocity of \(AB\) is \(\frac{u\sin(\alpha-\beta)}{a\cos\beta}\), where \(u\) is the velocity of \(A\).
Two particles of masses \(M, m\) are connected by an inextensible string, and lie on a smooth table with the string fully extended. A particle of mass \(m'\) impinges directly on \(m\) with velocity \(V\) in a direction making an acute angle \(\alpha\) with the string. Prove that the direction of the velocity of \(m\) after impact makes an angle \(\tan^{-1}\left(1+\frac{M}{m}\right)\tan\alpha\) with the string and that the velocity of \(M\) is \[ mm'V\cos\alpha/[m(m+m')+M(m+m'\sin^2\alpha)], \] assuming that \(m, m'\) are inelastic.
Two weights \(W, W'\) balance on any system of pullies with vertical strings. If a weight \(w\) be attached to \(W\), shew that it will descend with acceleration \[ g / \left[1 + \frac{W(W+W')}{wW'}\right], \] neglecting the inertia of the pullies.
A particle slides down the surface of a smooth fixed sphere of radius \(a\) starting from rest at the highest point. Find where it will leave the sphere, and shew that it will afterwards describe a parabola of latus rectum \(\frac{16}{27}a\), and that it will strike the horizontal plane through the lowest point of the sphere at a distance \(\frac{5(\sqrt{5}+4\sqrt{2})a}{27}\) from the vertical diameter.
A horizontal board is made to perform simple harmonic oscillations horizontally, moving to and fro through a distance 30 inches and making 15 complete oscillations per minute. Find the least value of the coefficient of friction in order that a heavy body placed on the board may not slip.
State and prove the rule for finding the highest common factor of two rational integral functions of \(x\). Find two rational integral functions \(X\) and \(X'\) of \(x\) such that \[ X(x^3+5x^2+7x+4) - X'(x^2+2x+3) = 1. \]
Solve the equation \[ \frac{1}{\sqrt{a+x}-\sqrt{a}} + \frac{1}{\sqrt{a+x}+\sqrt{a}} = \frac{m}{\sqrt{a+x}-\sqrt{a-x}}. \] If \begin{align*} ax^2 &= 1/y+1/z, \\ by^2 &= 1/z-1/x, \\ cz^2 &= 1/x+1/y, \end{align*} prove that \[ abcx^2y^2z^2 = b+c-a, \] hence solve the equations.
Find the conditions that
Prove that \[ \log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\dots \] when \(|x|<1\). Prove that \[ (1+x)^{1-x} = 1+x+x^2+\frac{1}{2}x^3+\frac{1}{3}x^4+\dots. \]