PYQs

1) Find the length of an endless chain which will hang over a circular pulley of radius ${}^{\prime }{a}^{\prime }$ 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 $\mu \log \left[{\displaystyle \frac{1+\sqrt{1+{\mu }^2}}{\mu }}\right]$, 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 ${}^{\prime }{a}^{\prime }$ 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 $\mathrm{c}\mathrm{m}$ and 26 $\mathrm{c}\mathrm{m}$. 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 $c\{{\displaystyle \frac{3}{\log (2+\sqrt{3})}}+{\displaystyle \frac{4\pi }{3}}\}$.

[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]