Question

Consider regular polygons with number of sides 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 slieeping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $△$ Then $△$ depends on n and h as

• A. $△=hsi{n}^2\frac{\pi }{n}$
• B. $△=h[\frac{1}{cos\left(\pi /n\right)}-1]$
• C. $△=hsin\left(\frac{2\pi }{n}\right)$
• D. $△=hta{n}^2\left(\frac{\pi }{2n}\right)$

Answer 1

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

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