If \(x\) and \(a\) are small and \(e^x \tan \frac{x}{2} = a\), prove by successive approximation that the first four terms in the expansion of \(x\) in powers of \(a\) give \(x = 2a - \frac{4}{3}a^2 + \frac{26}{9}a^3 - \frac{13}{3}a^4\).
State the principle of Virtual Work. Prove it (i) for forces acting at a point, (ii) for forces acting on a system of connected particles; and explain the application of the principle to the case of a rigid body under given forces, stating what forces may be omitted and the reasons for doing so. Deduce in the case of coplanar forces necessary and sufficient conditions of equilibrium.
How many tons of coal, having a calorific value of 8000 Thermal units per pound, would be required per day to give the same output from engines and boilers having an efficiency of \(9\%\), as would be obtained by utilizing the whole of the average flow of 270,000 cubic feet per second of water over Niagara Falls in turbines giving an efficiency of \(80\%\), the height of the Falls being 160 feet? (1 Thermal unit = 1400 ft. lbs.)
Shew that the general equation of a circle passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) is \[ (x-x_1)\left(x-x_2+\frac{\lambda}{x_2-x_1}\right) + (y-y_1)\left(y-y_2-\frac{\lambda}{y_2-y_1}\right) = 0; \] and that the circles corresponding to the values \(\lambda_1, \lambda_2\) of \(\lambda\) are orthogonal if \[ \lambda_1\lambda_2 + (x_1-x_2)^2(y_1-y_2)^2=0. \]
A gun can send a shot to a given height; prove that the area commanded on an inclined plane through the point of projection is an ellipse, with its focus, \(S\), at the gun. Shew that, if \(Q\) be any point within this area, the product of the two possible times of flight from \(S\) to \(Q\) is \(2SQ/g\).
A hill station \(C\) is observed from each of two stations \(A\) and \(B\) at the same level on the plain below, and the elevations of \(C\) above the horizontal are found to be \(37^\circ\) and \(45^\circ30'\) respectively, whilst the base \(AB\), which is 1 mile in length, subtends an angle of \(47^\circ\) at \(C\). Find to the nearest foot the error in the calculated height of \(C\) above the plain which would be caused by an error of \(30'\) in the measurement of the latter angle at \(C\).
Shew how to determine graphically the resultant of a system of given coplanar but non-concurrent forces. Indicate the principle of the method, and distinguish between the cases of (i) a force resultant, (ii) a couple resultant. Illustrate the method by determining graphically in position and magnitude the resultant of three forces of relative magnitudes 1, 2, 3 acting along the sides \(AB, BC, AC\) respectively of an equilateral triangle, the directions of the forces being indicated by the order of the letters.
A 12 ton tram starts from rest up an incline of 1 in 100, and when it reaches a speed of 6 miles an hour it is accelerating at 0.5 ft. per sec. per sec. Friction is estimated at 10 lbs. weight per ton, and the efficiency of the motors is \(60\%\). Find the current taken by the car if the voltage of supply be 550. (1 Horse-power = 746 watts.)
Shew that the parabolas \[ y^2-4ax=0, \quad y^2+4ax-8aty+8a^2t^2=0, \] are equal, have their axes parallel, and touch one another at the point \((at^2, 2at)\). Shew also that if the straight line \(x=t(y-a)\) cuts the second parabola in the real points \((\alpha+\alpha', \beta+\beta')\) and \((\alpha-\alpha', \beta-\beta')\), it cuts the first in the imaginary points \((\alpha+i\alpha', \beta+i\beta')\) and \((\alpha-i\alpha', \beta-i\beta')\).
Two inclined planes intersect in a horizontal line, and are inclined to the horizontal at angles \(\alpha, \beta\); a particle is projected from a point on the former (distant \(a\) from the valley line) at right angles to that plane, and strikes the other plane also at right angles. Shew that the velocity of projection must be \[ [2ga \sin^2\beta / \{\sin\alpha - \sin\beta\cos(\alpha+\beta)\}]^{\frac{1}{2}}. \]