WBJEE · Physics · Units and Dimensions

A spherical liquid drop is placed on a horizontal plane. A small disturbance causes the volume of the drop to oscillate. The time period of oscillation ( \(T\) ) of the liquid drop depends on radius \((n\) of the drop, density \((\rho)\) and surface tension (5) of the liquid. Which among the following will be a possible expression for \(T\) (where, \(k\) is a dimensionless constant)?

A \(k \sqrt{\frac{\rho r}{S}}\)
B \(k \sqrt{\frac{\rho^{2} r}{S}}\)
C \(k \sqrt{\frac{\rho r^{3}}{S}}\)
D \(k \sqrt{\frac{\rho r^{3}}{s^{2}}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(k \sqrt{\frac{\rho r^{3}}{S}}\)

Step-by-step Solution

Detailed explanation

According to the question. time period. \(T \alpha r^{a} p^{b} s^{c}\) \(r=k r^{a} p^{b} s^{c}\) Thus, putting dimension, we get Equating the dimensions of both sides, we get \([T]=[\mathrm{L}]^{a}\left[\mathrm{ML}^{-3}]^ \mathrm{b}\left[\mathrm{MT}^{-2} ]^c\right.\right.\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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