Catenary
Catenary
A catenary is the curve that an idealized hanging chain assumes under its own weight when supported only at the ends.
The catenary is described by the equation
\[y=\dfrac{e^{a x}+e^{-a x}}{2 a}=\dfrac{\cosh a x}{a}\], the lowest point being at \(\left(0, \dfrac{1}{a}\right)\)
PYQs
Catenary
1) Find the length of an endless chain which will hang over a circular pulley of radius \('a'\) so as to be in contact with the two-thirds of the circumference of the pulley.
[2015, 12M]
2) The end links of a uniform chain slide along a fixed rough horizontal rod. Prove that the ratio of the maximum span to the length of the chain is \(\mu \log \left[\dfrac{1+\sqrt{1+\mu^{2}}}{\mu}\right]\), where \(\mu\) is the coefficient of friction.
[2012, 20M]
3) Find the length of an endless chain which will hang over a circular pulley of radius \('a'\) so as to be in contact with the three-fourth of the circumference of the pulley.
[2009, 15M]
4) A uniform string of length one metre hangs over two smooth pegs \(\mathrm{P}\) at different heights. The parts which hang vertically are of lengths 34 \(\mathrm{cm}\) and 26 \(\mathrm{cm}\). Find the ratio in which the vertex of the catenary divides the whole string.
[2007, 12M]
5) Show that the length of an endless chain, which will hang over a circular pulley of radius \(c\) so as to be in contact with two-third circumference of the pulley is \(c\left\{\dfrac{3}{\log (2+\sqrt{3})}+\dfrac{4 \pi}{3}\right\}\).
[2006, 15M]
6) A non-uniform string hangs under gravity. Its cross-section at any point is inversely proportional to the tension at the point. Prove that the curve in which the string hangs is an arc of a parabola with its axis vertical.
[2004, 12M]
7) Obtain the equation of the curve in which a string hangs under gravity from two fixed points (not lying in a vertical line), when line mass density at each of its points varies as the radius of curvature of the curve.
[2002, 12M]