A light horizontal beam, freely jointed at \(O\), is supported and loaded as shewn. Determine the reactions at the supports. \centerline{\includegraphics[width=0.8\textwidth]{beam_diagram.png}} The diagram shows a beam with supports at points labeled 2 and 3. It is jointed at O. Loads of weight 1, 1, 1 are applied at distances a, 3a, 5a from the left end. The joint O is at 2a, support 2 is at the left end, support 3 is at 4a. Distances between points are all 'a'. Determine the greatest weight which does not destroy equilibrium no matter where it is placed on the beam.
Given that \[ x+iy = \coth \frac{1}{2}(\xi+i\eta), \] express \(x\) and \(y\) separately in real form in terms of \(\xi\) and \(\eta\), and shew that, when \(\xi < 0\), \(y\) can be expressed in a series \[ -2\{e^{\xi}\sin\eta + e^{2\xi}\sin 2\eta + e^{3\xi}\sin 3\eta + \dots\}. \] What is the corresponding series when \(\xi > 0\)?
Shew that \[ \frac{d^n e^{-x^2}}{dx^n} = (-1)^n e^{-x^2} \phi_n(x), \] where \(\phi_n(x)\) is a polynomial of degree \(n\) in which the coefficient of \(x^n\) is \(2^n\). Establish the relations \begin{align*} \phi_n'(x) &= 2n \phi_{n-1}(x), \\ \phi_n''(x) - 2x\phi_n'(x) + 2n\phi_n(x) &= 0. \end{align*}
A line is determined by the parametric equations \(x = a_0t + a_1\), \(y = b_0t + b_1\). The parameters \(t\) and \(t'\) of corresponding points of two ranges on this line are connected by the relation \[ att' + b(t+t') + c = 0. \] Show that there is a symmetrical \((1, 1)\) correspondence between points of the two ranges, and that there are two points of the first range which correspond to themselves in the second. Show further that these self-corresponding points harmonically separate every corresponding pair of points. Such a correspondence is called an involution. Show that any pencil of conics through four points cuts any line in pairs of points in involution, and hence show that to any such pencil belong two parabolas and that the directions of the asymptotes of each conic of the pencil are harmonically separated by the directions of the axes of these parabolas.
A uniform rod of length \(2l\) rests within a hollow sphere of radius \(a\) in a vertical plane through the centre of the sphere. The sphere is rough, the angle of friction being \(\phi\). Shew that, if \(l
If \[ I_p = \int_{-1}^1 (1-t^p)^p dt, \] prove that \[ I_p = \frac{2p}{2p+1}I_{p-1} \quad (p>0). \] Shew that, if \(n\) is a positive integer, \[ I_n I_{n+\frac{1}{2}} = \frac{\pi}{n+1}, \] and deduce that \[ \frac{\pi}{n+1} < I_n^2 < \frac{\pi}{n}. \]
Evaluate \[ \int_0^1 \sqrt{\frac{1-x}{1+x}}dx, \quad \int_0^\infty \frac{dx}{x^4+1}, \quad \int_0^\infty \frac{dx}{\cosh x + 1}. \]
\(P\) is the point \((h, k)\) of the parabola \(y^2=4ax\). The normals to the parabola at \(Q\) and \(R\) pass through \(P\). Prove that the area of the triangle \(PQR\) is \[ (h+a)\sqrt{(k^2-32a^2)}. \]
Shew that if a uniform heavy string has its ends fixed and hangs freely \[ y=c\cosh \frac{x}{c}, \quad s=c\sinh \frac{x}{c}. \] \(ACB\) is a telegraph wire, the straight line \(AB\) being horizontal and of length \(2l\), and \(C\) the middle point of the wire is at a distance \(h\) below \(AB\). Shew that the length of the wire is approximately \[ 2l+\frac{4}{3}\frac{h^2}{l} - \frac{28}{45}\frac{h^4}{l^3}. \]
A circle of radius \(b\) rolls on the outside of a fixed circle of radius \(a\), and a point carried by the rolling circle and distant \(c (< b)\) from its centre traces out a curve \(C\). Prove that \(C\) will or will not have points of inflexion according as \[ c > \frac{b^2}{a+b} \quad \text{or} \quad c \le \frac{b^2}{a+b}. \]