Assuming the formula \[ \sin\theta = \theta \left(1-\frac{\theta^2}{\pi^2}\right)\left(1-\frac{\theta^2}{2^2\pi^2}\right)\left(1-\frac{\theta^2}{3^2\pi^2}\right)\dots, \] and the expansions of \(\sin\theta\) and \(\cos\theta\) in powers of \(\theta\), prove that \begin{align*} &\left(1+\frac{x}{2}\right)\left(1-\frac{x}{3}\right)\left(1+\frac{x}{5}\right)\left(1-\frac{x}{7}\right)\left(1+\frac{x}{9}\right)\left(1-\frac{x}{11}\right)\dots \\ &= 1 + \frac{\pi}{4}x - \frac{\pi^2}{4^2}\frac{x^2}{2!} - \frac{\pi^3}{4^3}\frac{x^3}{3!} + \frac{\pi^4}{4^4}\frac{x^4}{4!} + \frac{\pi^5}{4^5}\frac{x^5}{5!} - \dots, \end{align*} a change of sign occurring after each two terms.
A 12 in. gun fires a projectile weighing 850 lbs., the travel of the latter in the bore being 32.25 ft. The curve connecting pressure and travel of projectile is as follows:
Define the eccentric angle of a point on an ellipse; and find the equation of the tangent and normal at any point in terms of the eccentric angle. Tangents \(TP, TQ\) are drawn to the ellipse \(x^2/a^2 + y^2/b^2=1\) from the point \(T\), \((x=a\cos\rho\sec\sigma, y=b\sin\rho\sec\sigma)\). Prove that the eccentric angles of \(P\) and \(Q\) are \(\rho \pm \sigma\), and that \[ TP^2 - TQ^2 = \pm (a^2-b^2)\tan^2\sigma \sin 2\rho \sin 2\sigma. \]
State and prove the theorem of conservation of linear momentum for a system of particles. Interpret the equations \[ v = u+ft, \quad v^2 = u^2+2fs, \] in terms of the impulse and work of a force. What do your statements reduce to in the case of an impulsive force? Two particles of masses \(m\) and \(m'\) are attached to the ends of an elastic string and placed at rest on a smooth horizontal table with the string just taut. The particle of mass \(m\) is given a velocity \(v\) outwards along the direction of the string. Find its velocity when the string is at its greatest extension and when it is next unstretched. Find equations determining the velocities of the particles, when the string has half its greatest extension.
A vertical iron door, 6 feet high, 4 feet broad and 1 inch thick, and weighing 490 pounds per cubic foot, is swinging to, its outer edge moving at 6 feet per second. Neglecting friction, find the least steady force which, applied at its outer edge, will stop it while it swings through 10 degrees.
From two points \((h, k), (h', k')\) tangents are drawn to the rectangular hyperbola \(xy=c^2\). Prove that the two points and the four points of contact will lie on a circle if \(hh' = kk'\), \(hk' + kh' = 4c^2\).
A pulley 3 ft. 6 ins. in diameter, making 150 revolutions a minute, drives by a belt a machine which absorbs 7 horse-power (1 horse power = 33000 ft. lbs. of work per minute). If the tension on the driving side is twice that on the slack side, and the maximum tension is to be 35 lbs. per inch width, find the width of the belt.
Prove that the equation \[ 7x^2 - 3xy + 3y^2 - 15x + 5y - 5 = 0 \] represents an ellipse whose minor axis passes through the origin. Find the coordinates of the centre and the lengths of the principal axes, and sketch the curve.
Define the hodograph. Shew that if \(P\) be a moving point and \(Q\) the corresponding point in the hodograph, the velocity of \(Q\) represents in magnitude and direction the acceleration of \(P\). Find the acceleration of a point moving uniformly in a circle. The weight of a truck is 10 tons and it is moving at a speed of 15 miles per hour round a curve of radius 4000 feet, which is banked up so that there shall be no lateral thrust on the rails at a speed of 20 miles per hour. Find the lateral thrust of the truck on the rails.
What is meant by the statement that ``the mechanical equivalent of a Thermal Unit in Pound-Centigrade units is 1400 ft. lbs.''? Describe any experiment for determining this ratio. A 30 H.P. petrol engine at full load consumes 21 lbs. of petrol per hour, the calorific value of the fuel being 9500 Pound-Centigrade Thermal Units per pound. Find the thermal efficiency of the engine. If the power is absorbed by a friction brake kept cool by a steady stream of water, which is supplied at 20\(^{\circ}\) C. and slowly boiled away, find how much water will be used per hour, if the latent heat of steam at atmospheric pressure be 536 Thermal Units.