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Question: Answered & Verified by Expert
If $\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{ML}^{3}}{3 \mathrm{Yq}}}$ then find the dimensions of q. Where $\mathrm{T}$ is the time period of bar of mass $\mathrm{M}$ length $\mathrm{L}$ and Young modulus $\mathrm{Y}$.
PhysicsUnits and DimensionsBITSATBITSAT 2011
Options:
  • A $[\mathrm{L}]$
  • B $\left[\mathrm{L}^{2}\right]$
  • C $\left[\mathrm{L}^{4}\right]$
  • D $\left[\mathrm{L}^{3}\right]$
Solution:
2492 Upvotes Verified Answer
The correct answer is: $\left[\mathrm{L}^{4}\right]$
$\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{ML}^{3}}{3 \mathrm{Yq}}},$ writing dimensions of both
the sides, we get $[\mathrm{T}]=\left[\frac{\mathrm{ML}^{3}}{\mathrm{ML}^{-1} \mathrm{~T}^{-2} \mathrm{q}}\right]^{1 / 2}$
or $\quad q=\left[L^{4}\right]$

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