A scale-pan weighing 1 lb. is attached to a light spiral spring and causes it to extend 2 inches. A 2 lb. weight is then placed in the pan and suddenly released. Find how far the pan will fall, the tension of the spring when the pan is at its lowest point, and the period of oscillation.
A body makes complete revolutions about a fixed horizontal axis, about which its radius of gyration is \(k\), and the centre of gravity of the body is at a distance \(c\) from the axis. If the greatest and least angular velocities are \(p\) per cent. greater and \(p\) per cent. less than a quantity \(\omega\), prove that \[ \omega = \sqrt{\frac{200gc}{k^2}}. \]
If a shot travelling with velocity \(v\) is subject to a retardation \(kv^3\) on account of air resistance, prove that (neglecting gravity) \[ 2kt = \frac{1}{v^2} - \frac{1}{u^2} \] and \[ kx = \frac{1}{v} - \frac{1}{u}, \] where \(u\) is the initial velocity, \(v\) the velocity after \(t\) seconds, and \(x\) the space described in that time.
A heavy lever (weight \(w\) lb. per foot length) with the fulcrum at one end, is to be used to raise a weight W, which is at a given distance \(a\) feet from that end. Prove that in order that the weight may be lifted with the least effort, the length of the lever should be \[ \sqrt{\frac{2aW}{w}} \text{ feet}. \]
Find the equation of the normal at any point \((at^3, at^2)\) of the curve \(x^2 = ay^3\), and show that it meets the curve again in two points whose parameters \(t_1, t_2\) are connected with \(t\) by the relation \[ \frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t}=0. \] (Note: The parameterization in the question appears to be inconsistent with the curve equation. Transcribed as written.) Prove that if \(t^2 = \frac{2}{9}\) the normal touches the curve.
Prove that the \(n\)th differential coefficient of \(e^{ax}\sin bx\) is \[ (a^2+b^2)^{\frac{n}{2}}e^{ax}\sin\left(bx+n\tan^{-1}\frac{b}{a}\right). \] Prove that the curve \[ y = e^{-x^2} \] has points of inflexion where \(x=\pm\frac{1}{\sqrt{2}}\). Sketch the curve roughly.
Prove that the line joining the vertex of a triangle to the point on the inscribed circle, which is furthest from the base, passes through the point of contact of the escribed circle with the base.
Given a circle of which AB is a diameter, C and D two points on the circumference, find a point P on the circumference such that PC and PD will cut AB in points equidistant from the centre.
Prove that the locus of the extremities of parallel diameters of a system of coaxal circles is a rectangular hyperbola.
Prove that, if \((1+x)^n = p_0+p_1x+\dots+p_nx^n\), where \(n\) is a positive integer, \[ \frac{p_0}{n+1} - \frac{1}{2}\frac{p_1}{n+2} + \dots + (-1)^r \frac{p_r}{n+r+1} + \dots = \frac{2^n(n!)^2}{(2n+1)!}. \]