Question

Consider regular polygons with number of side $n=3,4,5,....$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon as $△$. Then $△$ depends on $n$ and $h$ as

• A. $△=h{\sin }^2\frac{\pi }{n}$
• B. $△=h(\frac{1}{\cos \left(\frac{\pi }{n}\right)}-1)$
• C. $△=h\sin \left(\frac{2\pi }{n}\right)$
• D. $△=h{\tan }^2\left(\frac{\pi }{2n}\right)$

Answer 1

The correct option is B $△=h(\frac{1}{\cos \left(\frac{\pi }{n}\right)}-1)$

$\theta =\frac{\pi }{2}-\frac{\pi }{2}\sin \theta =\cos \frac{\pi }{n}l=\frac{h}{\sin \theta }△=l-h,=h[\frac{1}{\sin \theta }-1]=h[\frac{1}{\cos \frac{\pi }{n}}-1]$