Two small magnets have their masses and lengths in the ratio 1 : 2. the maximum torques experienced by them in a uniform magnetic field are the same. For small oscillations, the ratio of their time periods is

Options

A. `(1)/(2 sqrt(2))`
B. `(1)/(sqrt(2))`
C. `(1)/(2)`
D. `2 sqrt(2)`

Answer

a

Explanation

Let the mass of the first magnet = m
`:.` Mass of second magnet = 2 m
The time period of osillation of the magnet
`T = 2 pi sqrt((I)/(MB_(H)))`
Where, I = MI of magnet about the axis passing through it centre
`B_(H)` = magnetic moment of magnet
`:. (T_(1))/(T_(2)) = sqrt((I_(1))/(M_(1)) xx (M_(2))/(I_(2)))`
Let length of the magnet = L
MI of the magnets about the axis passing through one of the ends is `(mL^(3))/(3)`
`:. I_(1) = mL^(2)//3` and `I_(2) = ((2m)(2L)^(2))/(3)`
`(T_(1))/(T_(2)) = sqrt((mL^(2)//3)/(m_(1)) xx (M_(1))/((2m xx (2L)^(2))/(3)))`
It is given that maximum troques experienced by the magnets are same
`:. M. B_(H) = M_(2)B_(H) implies M_(1) = M_(2)` [ `:' m_(1) = M_(2)` (Given]
`implies (T_(1))/(T_(2)) = (1)/(sqrt(B)) = (1)/(2sqrt(2))`