Question

Consider regular polygons with number of sides $n=3,4,\mathrm{5.......}$ 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 is $\Delta$. Then $\Delta$ depends on $n$ and $h$ as

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

Answer 1

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

The answer C shows the each polygon depends on the n and h is given