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Question
In a certain unit, the radius of gyration of a uniform disc about its central and transverse axis is `sqrt2.5`. Its radius of gyration about a tangent in its plane (in the same unit) must be ______.
Options
`sqrt5`
2.5
`2sqrt2.5`
`sqrt12.5`
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Solution
In a certain unit, the radius of gyration of a uniform disc about its central and transverse axis is `sqrt2.5`. Its radius of gyration about a tangent in its plane (in the same unit) must be 2.5.
Explanation:
The expression for the M.I. of a uniform disc about its central and transverse axis is given by
M.I = `(m xx R^2)/2` ......(i)
Here, m and R are mass and radius of gyration, respectively.
Let K be the radius of gyration of the disc. Then, from (i) we can write,
M.I = `(m xx R^2)/2`
`m xx K^2 = (m xx R^2)/2`
`K = R/sqrt(2)`
Therefore, `sqrt2.5 = R/sqrt(2)`
`R = sqrt(5)` ......(ii)
Now, the expression for the moment of inertia of a uniform disc about a tangent is given by
M.I = `(5 m R^2)/4`
If K' be the radius of gyration of the disc about the tangent, then we can write
M.I = `(5 m R^2)/4`
m × (K')2 = `(5 m R^2)/4`
(K')2 = `(5 R^2)/4`
(K')2 = `(5(sqrt 5)^2)/4`
(K')2 = `(5 xx 5)/4`
(K')2 = `25/4`
(K') = 2.5
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