Q. A spherical body of radius \( r \) and density \( \sigma \) falls freely through a viscous liquid having density \( \rho \) and viscosity \( \eta \) and attains a terminal velocity \( v_0 \). Estimated maximum error in the quantity \( \eta \) is :
(Ignore errors associated with \( \sigma \), \( \rho \) and \( g \))
A. \( 2 \left[ \frac{\Delta r}{r} - \frac{\Delta v_0}{v_0} \right] \)
B. \( 2 \left[ \frac{\Delta r}{r} + \frac{\Delta v_0}{v_0} \right] \)
C. \( \frac{2\Delta r}{r} + \frac{\Delta v_0}{v_0} \)
D. \( 2\frac{\Delta r}{r} - \frac{\Delta v_0}{v_0} \)
Correct Answer: \( \frac{2\Delta r}{r} + \frac{\Delta v_0}{v_0} \)
Step 1: Use Stokes’ Law
Terminal velocity of sphere:
\[
v_0 = \frac{2 r^2 (\sigma - \rho) g}{9 \eta}
\]
Step 2: Rearranging for η
\[
\eta = \frac{2 r^2 (\sigma - \rho) g}{9 v_0}
\]
Step 3: Identify Variable Dependence
\[
\eta \propto \frac{r^2}{v_0}
\]
Step 4: Apply Error Propagation
For multiplication and division:
\[
\frac{\Delta \eta}{\eta} = 2\frac{\Delta r}{r} + \frac{\Delta v_0}{v_0}
\]
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