Prove that the square of the distance between the centres of the inscribed circle and the circumscribed circle of a triangle is \(R^2-2Rr\). Prove that the sum of the squares of the distances of the circumcentre from the inscribed and escribed centres is \(12R^2\).
If \(p\) and \(q\) are integers prime to each other prove that \((\cos\theta+i\sin\theta)^{p/q}\) has \(q\) different values, and find them. If \(\sin(x+iy) = \tan(u+iv)\), show that \(\sin 2u \tanh y = \sinh 2v \tan x\).
Find the conditions of equilibrium of a number of forces acting at given points in a plane. A uniform heavy wire is bent into the form of a triangle \(ABC\) and is suspended by a string attached to the middle point of \(BC\). Prove that in equilibrium \(AB\) and \(AC\) are equally inclined to the horizontal.
A series of \(n\) uniform rods \(A_0A_1, A_1A_2, \dots\) are freely jointed together and hang in a vertical plane with the ends of the extreme rods \(A_0, A_n\) attached to fixed points. Show how to obtain, graphically or otherwise, the actions at the joints. If the weights of the rods in order are in arithmetical progression, prove that the tangents of the angles which the rods make with the horizontal are also in arithmetical progression.
A heavy particle of weight \(W\) is to be supported by a given force equal to \(W/2\) on the upper portion of the outer surface of a fixed rough circular cylinder. If the coefficient of friction is equal to \(1/\sqrt{3}\) find the greatest angular distance from the highest generator of the cylinder at which the particle can be maintained in equilibrium.
Define work, energy, horse-power. Find the average horse-power of the engine required to pump out a dock 450 ft. long, 60 ft. broad and 30 ft. deep in three hours at a uniform rate, if the water is delivered at a level 40 ft. above the bottom of the dock through a 30 inch pipe (a cubic foot of water weighs 62.5 lbs.).
A particle is projected under gravity with a given velocity and in a given direction. Find equations to determine its direction of motion and position at any time. A particle is projected from a point at a height \(h\) above the ground so as to strike it at a horizontal distance \(c\) from the point of projection. Show that the least velocity of projection \(v\) is given by \[ v^2 = g\{\sqrt{c^2+h^2}-h\}. \]
A simple pendulum of length \(l\) swings through an angle \(\alpha\) to each side of the vertical. Find the tension of the string when it makes an angle \(\theta\) with the vertical. If \(\alpha\) is very small, show that the time of a semi-swing is \(\pi\sqrt{l/g}\). Assuming that \(g\) varies inversely as the square of the distance from the centre of the earth, find approximately the height of a mountain at the top of which a pendulum which beats seconds at sea level loses 20 seconds a day. The radius of the earth may be taken as 4000 miles.
A regular hexagon \(ABCDEF\) formed of light rods is suspended from \(A\) and stiffened by light rods \(FB, FC, FD\), all the rods being freely jointed. Weights, each equal to \(W\), are attached to the joints \(B, C, D, E, F\). Find graphically or otherwise the stress in each rod.
Prove that an inextensible string carrying a uniform load per unit horizontal length hangs in a parabola. A trolley-wire is carried on poles round a curve of 1200 feet radius. The poles are spaced 40 yards apart, and in the middle of each span the wire sags 6 inches below the points of support. If the wire weighs \(\frac{1}{2}\) lb. per foot, show that the resultant horizontal pull on each pole is very nearly 180 lb.