A boiler is fitted with a feed-water heater in the flue, which reduces the temperature of the flue-gases from \(350^\circ\) C. to \(200^\circ\) C., whilst the temperature of the feed-water is raised from \(40^\circ\) C. to \(84^\circ\) C. If the consumption of water and coal per hour be 1350 lbs. and 125 lbs. respectively, find the weight of air used per pound of coal, taking the specific heat of the flue-gases to be 0.24.
Find the coordinates of the limiting points of the system of coaxal circles of which \begin{align*} x^2 + y^2 - 2x - 6y + 2 &= 0, \\ 3x^2 + 3y^2 + 6x - 12y + 14 &= 0 \end{align*} are members.
A ring of mass \(m\) slides on a smooth vertical rod; attached to the ring is a light string passing over a smooth peg distant \(a\) from the rod, and at the other end of the string is a mass \(M(>m)\). The ring is held on a level with the peg and released: shew that it first comes to rest after falling a distance \[ \frac{2mMa}{M^2-m^2}. \]
Show that \[ \log_{10} 317 = 1 + 5\log_{10}2 + \log_{10}(1-\tfrac{3}{320}). \] Given that \(\log_{10}e = \cdot4343\dots\) and \(\log_{10}2 = \cdot3010\dots\), calculate \(\log_{10}317\) to 3 places of decimals.
Solution: \begin{align*} \log_{10}317 &= \log_{10}\left(320 \cdot \left( \frac{317}{320}\right)\right) \\ &= 1+5\log_{10}2 + \log_{10}\left(1-\frac{3}{320}\right)\\ &= 1+5\cdot0.3010+ \log_{10}\left(1-\frac{1}{106.666\ldots}\right)\\ &\approx 1+1.5050-\frac1{106}\log_{10}e\\ &=2.5050-0.0043\\ &=2.501 \end{align*}
Discuss the applications of the principles of energy and linear momentum to the solution of dynamical problems. \par A light smooth pulley rests on a smooth horizontal table. A mass \(4m\) is attached to it by a string, and a string passing round the pulley is attached to masses \(3m\) and \(m\). The strings are all parallel, and the smaller masses lie in the opposite side of the pulley to the mass \(4m\). The mass \(4m\) is set into motion by a given impulse applied to it in the direction of the string attached to it. Find the initial and subsequent motion of the system.
Prove that if the internal energy of a certain gas is a function of the temperature only, and its pressure, specific volume and temperature satisfy the equation \(pv = Rt\), where \(R\) is a constant, then the difference between the specific heat at constant pressure and the specific heat at constant volume, measured in work units, is equal to \(R\).
Prove that the general equation of a conic whose centre is the origin and which cuts the lines \(x=a\), \(y=\beta\) at right angles is \[ \frac{x^2}{\alpha^2} + 2\lambda xy + \frac{y^2}{\beta^2} + \lambda\left(1 - \frac{x^2}{\alpha^2} - \frac{y^2}{\beta^2}\right) = 0. \]
A mine cage, weighing with its load 5 cwt., is raised by an engine which exerts a constant turning moment on the rope drum which is 16 ft. in diameter. The speed rises until the engine is running at 60 revolutions per minute, when its output is 55 horse-power. \par Find the acceleration and the time that elapses before the cage reaches full speed: also find how far the cage rises in that time.
Find the equation of the circle which passes through the origin, has its centre on the line \(x+y=0\), and cuts the circle \[ x^2+y^2-4x+2y+4=0 \] at right angles.
Give a discussion of the hodograph and its applications. Shew that the motion of a moving point is completely given when its path and the hodograph of its motion constructed with a given pole are given. Illustrate your arguments by reference to the hodograph of a point on the rim of a wheel rolling with uniform velocity along a level road. \par A point describes a circle of radius \(a\) so that its hodograph is a second circle of radius \(b\). If the pole of the hodograph be at distance \(c\) from its centre, where \(c/b\) is small, shew that the time of a complete revolution is approximately \[ 2\pi a(1+\tfrac{1}{2}c^2/b^2)/b. \]