A cone with height (h= 1 m ) and a base circle of radius ( r = 1 m) is formed from a sector- shaped sheet of paper. The sheet is of such a size and shape that its two straight edges almost touch on the sloping surface of the cone. In this state the cone is stress-free. The cone is placed on a horizontal, slippery table-top, and loaded at its apex with a vertical force of magnitude $w = 2\pi$, without collapsing. The splaying of the cone is opposed by a pair of forces of magnitude $(F)$ acting tangentially at the join in the base circle (see figure). Ignoring any frictional or bending effects in the paper, find the value of $F$

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I tried to apply work energy theorem, and got $F\cdot \mathrm{d}r=w\cdot \mathrm{d}h$ but unsuccessful in proceeding further. any help will be much appreciated.

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    $\begingroup$ Homework type questions are discouraged on this site . Post your working as well... $\endgroup$ – Blazar Nov 21 '18 at 13:52

First of all trust me don't read this problem it is fairly easy. We start to attempt the solution by realising the fact that if the load depresses the cone's Apex then it would also increase the base radius or perimeter by a small amount let these decrease and increase be $h$ and $r$ respectively. So using work energy theorem we get $w(h) = f(2πr)$ as there is no other energy change. Now we must realise the most essential part of the solution that is the changes in the perimeter and height are connected as the base radius $R$ and height $H$ always obey $H^2+R^2= \rm constant$, differentiate this result to get $\frac{r}{h} = \frac{H}{R}$ and you finally have solved the problem!

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  • $\begingroup$ thanks. i was not able to think about the relation between H and r. that was the tricky part in problem. $\endgroup$ – Be happy Nov 22 '18 at 10:58

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