Shew that there is one point at which a rigid body can be supported so that it will be in equilibrium in all positions under the influence of gravity. Find the position of the centre of gravity of a uniform piece of wire bent into the form of a semi-circle with its diameter.
The infinite series \begin{equation} c_0 + c_1 + \dots + c_n + \dots \tag{1} \end{equation} and the infinite continued fraction \begin{equation} b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \dots + \cfrac{a_n}{b_n + \dots}}} \tag{2} \end{equation} are said to be equivalent if, for each value of \(n\), \[ S_n = \frac{p_n}{q_n}, \] where \begin{align*} S_n &= c_0 + c_1 + \dots + c_n, \\ \frac{p_n}{q_n} &= b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \dots + \cfrac{a_n}{b_n}}}. \end{align*} By finding the value of \(\frac{p_n}{q_n} - \frac{p_{n-1}}{q_{n-1}}\), or otherwise, shew that the continued fraction (2) is equivalent to the series \[ b_0 + \frac{a_1}{q_1} - \frac{a_1a_2}{q_1q_2} + \dots + (-1)^{n-1} \frac{a_1a_2\dots a_n}{q_{n-1}q_n} + \dots. \] Again, by solving the equations \begin{align*} s_n &= b_n s_{n-1} + a_n s_{n-2}, \\ 1 &= b_n \cdot 1 + a_n \cdot 1, \end{align*} for \(a_n\) and \(b_n\), or otherwise, shew that the series (1) is equivalent to the continued fraction \[ c_0 + \cfrac{c_1}{1 - \cfrac{c_2/c_1}{1 + c_2/c_1 - \cfrac{c_3/c_2}{1+c_3/c_2 - \dots - \cfrac{c_n/c_{n-1}}{1+c_n/c_{n-1} - \dots}}}} \] and find the series equivalent to \[ a_0 + \cfrac{\alpha_1}{1 - \cfrac{\alpha_2}{1+\alpha_2 - \dots - \cfrac{\alpha_n}{1+\alpha_n - \dots}}}. \]
Sum the series \[ 1^3 + 3^3 + 5^3 + \dots + (2n-1)^3. \]
Prove that, if \(a\) and \(b\) are positive integers (\(a < b\)), the proper fraction \(a/b\) can be expressed as a terminating series \[ \frac{1}{q_1} + \frac{1}{q_1q_2} + \frac{1}{q_1q_2q_3} + \dots, \] where \(q_1, q_2, q_3, \dots\) are positive integers in ascending order. Illustrate the process by the fraction \(12/29\).
A uniform plank 16 feet long is supported horizontally at two points distant 4 feet from the ends. Draw two sets of diagrams to represent the shearing force and bending moment in the plank when a heavy particle whose weight is one half that of the plank is placed on it (a) at one end, (b) at the middle.
A point \(P\) divides \(AB\) in the ratio \(\lambda : \mu\); \(x, x_1, x_2\) are the distances (measured in a fixed direction) of \(P, A, B\) from a given straight line; prove that \(x = (\mu x_1 + \lambda x_2)/(\lambda+\mu)\). Obtain an equation giving the ratios in which \(AB\) is divided by a conic whose equation (in rectangular or oblique coordinates) is \(\phi(x,y)=0\), and deduce the equations (1) of a tangent at a point on the curve, (2) of the tangents from a point not on the curve, (3) of the polar of any point. Apply the method to find the double and multiple points (if any) on an algebraic curve, and the tangents at such a point. Extend the method to equations in homogeneous point coordinates, such as areals or trilinears.
Obtain by a geometrical construction, or otherwise, the solutions of the equations \begin{align*} 5 \sin \theta - 2 \sin \phi &= 1, \\ 5 \cos \theta - 2 \cos \phi &= 4, \end{align*} which lie between \(\pm 180^\circ\).
Prove that, if \begin{align*} \sin(\beta+\gamma) + \sin(\gamma+\alpha) + \sin(\alpha+\beta) &= 0 \\ \text{and} \quad \alpha+\beta+\gamma &= \theta, \\ \text{then} \quad \frac{\sin\alpha+\sin\beta+\sin\gamma}{\sin\theta} = \frac{\cos\alpha+\cos\beta+\cos\gamma}{\cos\theta} &= \cos(\beta+\gamma) + \cos(\gamma+\alpha) + \cos(\alpha+\beta). \end{align*}
A uniform rod of weight \(W\) and length \(l\), lies on a rough horizontal plane, the coefficient of friction being \(\mu\). A string is attached to one end, and is pulled horizontally in a direction perpendicular to the rod so that the tension gradually increases. Shew that the rod begins to turn about a point \(\frac{l}{\sqrt{2}}\) from the end to which the string is attached, and that the tension of the string is then \((\sqrt{2}-1)\mu W\), assuming that the vertical reaction is distributed uniformly along the rod.
A function \(f(x)\) and as many of its derivatives as are required are single valued and continuous for values of \(x\) in the neighbourhood of a given value \(a\). If \quad \(\psi(x) = f(a+h) - f(a+h-x) - xf'(a+h-x) - \dots - \frac{x^n}{n!}f^{(n)}(a+h-x)\), shew that \quad \(f(a+h) = f(a) + hf'(a) + \dots + \frac{h^n}{n!}f^{(n)}(a) + \psi(h)\), so that \(\psi(h)\) is the remainder in Taylor's theorem. Prove that \quad \(\psi'(x) = \frac{x^n}{n!} f^{(n+1)}(a+h-x)\), and deduce \quad \(\psi(h) = \frac{1}{n!} \int_0^h x^n f^{(n+1)}(a+h-x)\,dx\). Obtain this in the form \[ \frac{h^{n+1}}{n!} \int_0^1 f^{(n+1)}(a+th)(1-t)^n\,dt, \] and discuss its behaviour as \(n \to \infty\), when