Prove that \[ \sum_{r=0}^{n-1} \frac{1}{1-\cos\left(\phi+\frac{2r\pi}{n}\right)} = \frac{n^2}{1-\cos n\phi}. \]
A body of mass one lb. is projected on a rough plane surface with a velocity of 10 feet per second, and its velocity after time \(t\) is given for various values of \(t\) by a smooth curve passing through the points defined by the following table:
Prove that the area bounded by the hyperbola \(xy=1\), the axis of \(x\), and the ordinates \(x=1\) and \(x=2\), is less than 1; and that similarly bounded by the ordinates \(x=1\) and \(x=3\) is greater than 1.
Give an account of the theory of the parabolic motion of a projectile under the influence of gravity only. In particular describe the use of graphical methods depending on the geometrical properties of the focus and directrix of the path. Prove that the paths of all particles projected with a given velocity from a given point, the direction being arbitrary, touch a surface formed by the rotation of a certain parabola about the vertical line through the point of projection.
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 \(R\).