| When two bar magnets have their like poles tied together, they make 12 oscillations per minute and when their unlike poles are tied together, they make 4 oscillations per minute. |
| Find the ratio of their magnetic moments. |
A) \[\frac{2}{3}\]
B) \[\frac{3}{2}\]
C) \[\frac{5}{4}\]
D) \[\frac{4}{5}\]
Correct Answer: C
Solution :
[c] : When the like poles are tied together, the net magnetic moment is \[({{m}_{1}}+{{m}_{2}})\]and the moment of inertia is \[({{I}_{1}}+{{I}_{2}})\] \[\therefore \]The time period \[{{T}_{1}}=2\pi \sqrt{\frac{{{I}_{1}}+{{I}_{2}}}{({{m}_{1}}+{{m}_{2}})B}}.\] When the unlike poles are tied together, the net magnetic moment is \[({{m}_{1}}-{{m}_{2}})\], while the moment of inertia (being a scalar quantity) remains unchanged. \[\therefore \]The time period \[{{T}_{2}}=2\pi \sqrt{\frac{{{I}_{1}}+{{I}_{2}}}{({{m}_{1}}+{{m}_{2}})B}}.\] Thus, \[\frac{T_{2}^{2}}{T_{1}^{2}}=\frac{({{m}_{1}}+{{m}_{2}})}{({{m}_{1}}-{{m}_{2}})}\Rightarrow \frac{{{m}_{1}}}{{{m}_{2}}}=\frac{T_{2}^{2}+T_{1}^{2}}{T_{2}^{2}-T_{1}^{2}}=\frac{\upsilon _{1}^{2}+\upsilon _{2}^{2}}{\upsilon _{1}^{2}-\upsilon _{2}^{2}}\]. Given, \[{{\upsilon }_{1}}=12\]per minute and \[{{\upsilon }_{2}}=4\] per minute. \[\therefore \] \[\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{{{12}^{2}}+{{4}^{2}}}{{{12}^{2}}-{{4}^{2}}}=\frac{144+16}{144-16}=\frac{160}{128}=\frac{5}{4}\].
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