A uniform chain of length \(2l\) is hung between two points at the same level distant \(2b\) apart. Find the equation of the curve, and the vertical and horizontal components of the tension at any point.
A mass \(M\) rests on a smooth table and is attached by two inelastic strings to masses \(m, m'\), (\(m' > m\)), which hang over smooth pulleys at opposite edges of the table. The mass \(m'\) falls a distance \(x\) from rest, and then comes into contact with the floor (supposed inelastic). Show that \(m\) will continue to ascend through a distance \(y\) given by \[ y = \frac{(m'-m)(M+m)}{m(M+m+m')}x. \] Show further that when \(m'\) is jerked into motion again as \(m\) falls it will ascend a distance \[ \frac{(M+m)^2}{(M+m+m')^2}x. \]
The external resistance to the motion of a bicycle consists of a constant force together with a force proportional to the square of the velocity. The speed is constant at 10 ft. sec. when free wheeling down a slope of 1 in 50, and constant at 20 ft. sec. on a slope of 1 in 25. The mass of the bicycle and the rider is 200 lbs. Find the power expended in maintaining a steady speed of 15 ft. sec. on the level, assuming that 15 per cent. of the work is lost in friction in the pedalling gear.
A body is projected from the ground with velocity \(V\) at inclination \(\alpha\) to the horizontal. At the highest point of the trajectory the body is broken into two parts by an internal explosion which creates \(E\) ft. lbs. of energy without altering the direction of motion. Find the distance between the parts when they reach the ground.
A smooth straight tube rotates in a horizontal plane about a point in itself with uniform angular velocity \(\omega\). At time \(t=0\) a particle is placed inside the tube, at rest relative to the tube and distant \(b\) from the point of rotation. Show that at time \(t\) the distance of the particle from the point of rotation is \(b \cosh \omega t\).
Shew that the product \((a^3+b^3+c^3-3abc)(x^3+y^3+z^3-3xyz)\) can be expressed in the form \(A^3+B^3+C^3-3ABC\), all the quantities involved being real. Find \(A, B, C\) in terms of \(a,b,c,x,y,z\). Shew also that if \(n\) is a positive integer of the form \(3m+1\) (\(m\) being any positive integer), then \((y-z)^n + (z-x)^n + (x-y)^n\) has a factor \(\Sigma a^2 - \Sigma yz\).
(i) Find the sum to \(n\) terms of the series: \(1 + 2^2x + 3^2x^2 + \dots\). (ii) Find the sum of the infinite series: \(1+3x+\frac{5x^2}{2!} + \frac{7x^3}{3!} + \dots\).
(i) Solve \[ \frac{x^2-a^2}{(x-a)^3} - \frac{x^2-b^2}{(x-b)^3} + \frac{x^2-c^2}{(x-c)^3} = 0 \] \[ \frac{(x+a)^3}{(x+a)^3} - \frac{(x+b)^3}{(x+b)^3} + \frac{(x+c)^3}{(x+c)^3} = 0 \] for \(x\), where \(a,b,c\) are unequal. [Note: The second equation appears to have typos in the source; it's transcribed as written, but likely intended to be different.] (ii) Shew that if the roots of the equation \(ax^4+bx^3+cx^2+dx+e=0\) are in harmonic progression, then \(d^3=4cde-8be^2\), and \(25ad^2e = (cd-eb)(11eb-cd)\). Verify these conditions in the case of \(40x^4-22x^3-21x^2+2x+1=0\) and solve for \(x\).
Shew that the number of distinct sets of three positive integers (none zero) whose sum is the odd integer \(2n+1\), is given by the least integer containing \(\frac{n^2+n}{3}\).
(a) Differentiate with respect to \(x\): (i) \(x^{x^{\cosh^{-1}x}}\); (ii) \(\tan^{-1}\left[\tan x \frac{1+\cos 2x}{1-\cos 2x}\right]\). (b) Expand \(y=\tan^{-1}x\) as a series in increasing powers of \(x\), stating any theorems you may use, or conditions you may apply to \(x\).