Two equal ladders are hinged at the top and rest on a rough floor forming an isosceles triangle with the floor of vertical angle \(2\theta\). A man whose weight is \(n\) times that of either ladder goes slowly up one of them. Calculate the reactions at the floor when his distance from the top is \(x\), and show that slipping begins when \[ nx/l=(2\mu-\tan\theta)/(\mu-\tan\theta). \]
A frame, formed of four light rods of equal length, freely jointed at \(A,B,C,D\), is suspended at \(A\). A particle of mass \(m\) is suspended from \(B\) and \(D\) by two strings each of length \(l\). The frame is prevented from collapsing by a string \(AC\). Show that the tension of the string is equal to \(\frac{1}{2}mg\frac{AP}{PN}\), where \(P\) is the particle and \(N\) is the centre of the rhombus \(ABCD\).
\(AB\) is the horizontal diameter of a circular wire whose plane is vertical. A bead of mass \(M\) at the lowest point \(C\) can slide on the wire and is attached to two strings which pass through small fixed rings at \(A,B\). To the other ends of the strings are attached equal particles \(m\) which hang freely. Find the potential energy of the system when it is displaced so that the radius to \(M\) makes an angle \(\theta\) with the vertical. Deduce that the equilibrium with \(M\) at \(C\) is stable if \(m
Obtain the equation \(y=c\cosh\frac{x}{c}\) for the curve of a uniform chain hanging under gravity. If the chain is suspended from two points \(A,B\) on the same level and the depth of the middle below \(AB\) is \(l/n\), where \(2l\) is the length of the chain, show that the horizontal span \(AB\) is equal to \(l\left(n-\frac{1}{n}\right)\log\left(\frac{n+1}{n-1}\right)\). Find an approximation to the difference between the arc \(AB\) and the span \(AB\) when \(n\) is large.
A load \(W\) is to be raised by a rope, from rest to rest, through a height \(h\); the greatest tension which the rope can safely bear is \(nW\). Show that the least time in which the ascent can be done is \(\left\{\frac{2nh}{(n-1)g}\right\}^{\frac{1}{2}}\).
Find the horse power required to lift 1000 gallons of water per minute from a canal 20 feet below and project it from a nozzle of cross section 2 sq. inches. [1 c. foot of water weighs 62\(\frac{1}{2}\) lb. and 1 gallon of water weighs 10 lb.]
A pile-driver weighing 200 lb. falls through 5 feet and drives a pile which weighs 600 lb. through a distance of 3 inches. Find the average resistance offered to the motion of the pile, assuming that the two remain in contact after the blow. How many foot pounds of energy are dissipated during the blow?
A smooth sphere impinging on another one at rest; after the collision their directions of motion are at right angles. Show that if they are assumed perfectly elastic, their masses must be equal.
A particle is suspended from a fixed point by a light elastic string. Show that the period of vertical oscillation is that of a simple pendulum of length \(l-l_0\), where \(l\) is the equilibrium length of the string and \(l_0\) its natural length. If the oscillations are of amplitude \(a\) and if when the particle is at the lowest point of its path it receives a downward blow which gives it a velocity \(u\), show that the time from the lowest to the highest point of the new path is \[ \sqrt{\frac{l-l_0}{g}} + \sqrt{\frac{l-l_0}{g}}\left\{\pi-\tan^{-1}\left(u\sqrt{\frac{l-l_0}{a^2g}}\right)\right\}. \]
Any point \(X\) is taken in the side \(CD\) of a rectangle \(ABCD\), and the line through \(A\) perpendicular to \(AX\) cuts \(BC\) in \(Y\). Prove that, if \(XY\) cuts \(BD\) in \(N\), then \(AN\) is perpendicular to \(XY\).