Rotational Motion Question 53
Question: The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is:
[AIPMT (S) 2004]
Options:
A) 2 : 3
B) 2 : 1
C) $ \sqrt{5}:\sqrt{6} $
D) $ 1:\sqrt{2} $
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Answer:
Correct Answer: C
Solution:
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Key Idea: If a body has mass M and radius of gyration is K, then $ I=MK^{2} $ . Moment of inertia of a disc and circular ring about a tangential axis in their planes are respectively.
$ I _{d}=\frac{5}{4}M _{d}R^{2} $ $ I _{r}=\frac{3}{2}M _{r}R^{2} $
but $ I=MK^{2} $
$ \Rightarrow $ $ K=\sqrt{\frac{1}{M}} $
$ \frac{K _{d}}{K _{r}}=\sqrt{\frac{I _{d}}{I _{r}}\times \frac{M _{r}}{M _{d}}} $
or $ \frac{I _{d}}{I _{r}}\sqrt{\frac{(5/4)M _{d}R^{2}}{(3/2)M _{r}R^{2}}\times \frac{M _{r}}{M _{d}}}=\sqrt{\frac{5}{6}} $
$ \therefore $ $ I _{d}:I _{r}=\sqrt{5}:\sqrt{6} $
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