Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R 
Assertion A : A spherical body of radius (5 ± 0.1) mm having a particular density is falling through a liquid of constant density. The percentage error in the calculation of its terminal velocity is 4%. 
Reason R : The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius. 

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is NOT the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The Correct Option is C

To solve this question, we must examine both the assertion and the reason given:

• Assertion A: A spherical body of radius \( (5 \pm 0.1) \, \text{mm} \) having a particular density is falling through a liquid of constant density. The percentage error in calculating its terminal velocity is 4%.
• Reason R: The terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius.

First, let us consider the formula for terminal velocity (\( v_t \)) of a spherical object moving through a liquid, which is given by:

v_t = \frac{2}{9} \cdot \frac{r^2 (\rho_s - \rho_l) g}{\eta}

where:

• r is the radius of the sphere,
• \rho_s is the density of the sphere,
• \rho_l is the density of the liquid,
• g is the acceleration due to gravity,
• \eta is the viscosity of the liquid.

From the formula, we observe that the terminal velocity is directly proportional to the square of the radius of the sphere (\( r^2 \)). Therefore, Reason R, which states that the terminal velocity is inversely proportional to the radius, is incorrect.

Now, let's verify the assertion:

The percentage error in a measured quantity can be given as:

\left( \frac{\Delta v_t}{v_t} \right) \times 100\%

Using the relation for terminal velocity, the percentage error of a quantity that is proportional to \( r^2 \) is twice the percentage error in the radius (since \(\Delta (%r) = \frac{0.1}{5} \times 100 = 2\%\)). So, the percentage error in terminal velocity can be calculated as:

2 \times 2\% = 4\%

Therefore, Assertion A is true because the calculated 4% error in terminal velocity matches the statement in the assertion.

In conclusion, the correct choice is: A is true but R is false.