MHT CET · Physics · Mechanical Properties of Fluids

A spherical solid ball of volume \(V\) is made up of material of density \(\rho\). It is falling through a liquid of density \(\sigma(\sigma<\rho)\). Assume that the liquid applies a viscous force on the ball that is proportional to square of the terminal speed \(v_{\mathrm{T}}, F=-K v_{\mathrm{T}}^2, \forall(K>0)\), then the terminal speed of the ball is ( \(g=\) acceleration due to gravity)

A \(\left[\frac{V g \rho}{K}\right]^{\frac{1}{2}}\)
B \(\left[\frac{V g(\rho-\sigma)}{K}\right]^{\frac{1}{2}}\)
C \(\frac{V g(\rho-\sigma)}{K}\)
D \(\frac{V g \rho}{K}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\left[\frac{V g(\rho-\sigma)}{K}\right]^{\frac{1}{2}}\)

Step-by-step Solution

Detailed explanation

The condition for terminal speed \(v_{\mathrm{T}}\) is given by,
weight \((W)=\) buoyant force \((f)+\) viscous force \((F)\)
We know, \(W=\rho V g, f=\sigma V g\) and \(F=K v_{\mathrm{T}}^2\)
\(\therefore \rho V g=\sigma V g+K v_{\mathrm{T}}^2\)
\(v_{\mathrm{T}}=\sqrt{\frac{(\rho-\sigma) V g}{K}}\)

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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