19what is this polar M.I ?????
A wire of length l is bent into shape of an n sided regular polygon .If its mass be m , determine its
polar M.I. about its centroid.
Also find the M.I., in the limiting case as n-->∞
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4 Answers
62The total lenght is L
distance of the side from the center is (L/2n)cot (2pi/2n)
divide it into n parts.. each part has MI of
\\\frac{m(L/n)^2}{12}+m(\frac{L}{2n}\times cot(\frac{2\pi}{2n}))^2 \\=\frac{mL^2}{12n^2}[1+\frac{3}{tan^2(\frac{\pi}{n})}]
because each one has mass m, the final answer will also have the same expression, m gets replaced by M.
\frac{ML^2}{12n^2}[1+\frac{3}{tan^2(\frac{\pi}{n})}]
Limit n-> infintiy, the expression will tend to ...
\frac{ML^2}{12n^2}[1+\frac{3}{(\frac{\pi}{n})^2}]=\frac{ML^2}{12n^2}[1+\frac{3n^2}{\pi^2}]=\frac{ML^2}{4\pi^2}=\frac{M(2\pi r)^2}{4\pi^2}=Mr^2
I have used a lot of equalities above. but they are all with limits which i have not written..
1in translation motion the direction of linear momentum and velocity are same in all cases
but in rotational motion about fix axis the direction of angular momentum and angular velocity are same if body is symetric and their direction are diffrent if body is not symetric.
the above statement is true or false.
[source NCERT]
