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Question
Choose the correct option:
A rectangular magnet suspended freely has a period of oscillation equal to T. Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely. Its period of oscillation is T′, the ratio of T′/T is ______.
Options
`1/(2sqrt2)`
`1/2`
2
`1/4`
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Solution
A rectangular magnet suspended freely has a period of oscillation equal to T. Now it is broken into two equal halves (each having half of the original length) and one piece is made to oscillate freely. Its period of oscillation is T′, the ratio of T′/T is `1/2`.
Explanation:
Calculate time period (T) of the complete magnet:
T = `2pi sqrt(I/(mu * B)`
Calculate moment of inertia (I) of the complete magnet:
`I = 1/12 ml^2` .....(1)
Calculate time period (T) of the broken magnet
T' = `2pi sqrt((I"'")/(mu"'" * B)` ......(2)
Calculate moment of inertia (I') of the broken magnet
`I = 1/12 * m/2 * (l/2)^2 = 1/12 (ml^2)/8`
From equation (1):
`I"'" = I/8` also `mu = mu/2`
Substitute moment of inertia (1) in (2):
T' = `2pi sqrt((I"'")/(mu"'" * B)) = 2pi sqrt(I/(8 * mu"'" * B)`
Calaculate ratio `T/T`:
`T/T = (2pisqrt(I/(8 * mu/2 * B)))/(2pi sqrt(I/(mu*B))`
`T/T = 1/2`.
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