Deduce the equations of equilibrium for a uniform freely suspended string, shewing that the string hangs in the form of a catenary. Prove that the vertical tension at any point is equal to \(w\lambda\), where \(w\) is the weight of unit length of string, and \(\lambda\) is the vertical distance from the given point to the directrix of the catenary. A heavy string of length \(2l\) is hung from two fixed points \(A, B\) in the same horizontal line, at a distance apart equal to \(2a\). A weight \(W\) is attached to a certain point of the string. Shew that the parameters of the two catenaries in which the string hangs are the same, and shew that if \(W\) is in the middle of the string and its weight is great in comparison to that of the string, the parameter \(c\) is equal to \(Wa/2w\sqrt{l^2-a^2}\) nearly, while if on the other hand the weight of \(W\) is small in comparison with that of the string, \[ l=c\sinh\frac{a}{c} + \frac{W}{2w}\left\{\cosh\frac{a}{c}-1\right\}. \]
The springs of a motor car are such that the weight of the parts carried on the springs depresses the latter through 2 inches from the position when unloaded. Find the natural period in which the car bounces; and shew that, on a road in which there is a series of ridges at intervals of 6 ft., the bouncing may become excessive at a speed of about 9 miles an hour.
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.