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प्रश्न
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 ______.
विकल्प
`sqrt5`
2.5
`2sqrt2.5`
`sqrt12.5`
उत्तर
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:
Expression for 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 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, Expression for moment of inertia of uniform disc about tangent is given by:
M.I = `(5mR^2)/4`
If K' be the radius of gyration of disc about tangent then we can write:
M.I = `(5mR^2)/4`
m × (K')2 = `(5mR^2)/4`
(K')2 = `(5R^2)/4`
(K')2 = `(5(sqrt5)^2)/4`
(K')2 = `(5 xx 5)/4`
(K')2 = `25/4`
(K')2 = 2.5
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