Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : spherical body of radius`(5pm0.1)` mm having a particular density is falling through a liquid of constant density. The percentage error in the calulation 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.
In the light of the above statements, choose the correct answer from the options given below

Options

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

Answer

To solve this problem, we need to analyze both the Assertion (A) and the Reason (R) provided in the question.

### Step 1: Analyze Assertion A
The assertion states that a spherical body of radius \(5 \pm 0.1\) mm is falling through a liquid of constant density, and the percentage error in the calculation of its terminal velocity is \(4\%\).

The terminal velocity \(v_t\) of a sphere falling through a viscous fluid can be expressed by Stokes' law as:
\[ v_t = \frac{2}{9} \frac{r^2 (\rho_s - \rho_f) g}{\eta} \]
where:
- \(r\) is the radius of the sphere,
- \(\rho_s\) is the density of the sphere,
- \(\rho_f\) is the density of the fluid,
- \(g\) is the acceleration due to gravity,
- \(\eta\) is the viscosity of the fluid.

From this equation, we can see that the terminal velocity is proportional to the square of the radius (\(v_t \propto r^2\)).

### Step 2: Calculate the percentage error in terminal velocity
Given that the radius \(r = 5\) mm and the uncertainty in radius \(\Delta r = 0.1\) mm, we can calculate the percentage error in the terminal velocity using the relationship derived from the proportionality:
\[ \frac{\Delta v_t}{v_t} = 2 \frac{\Delta r}{r} \]
Substituting the values:
\[ \frac{\Delta v_t}{v_t} = 2 \frac{0.1 \text{ mm}}{5 \text{ mm}} = 2 \times 0.02 = 0.04 \]
To express this as a percentage:
\[ \text{Percentage error} = 0.04 \times 100\% = 4\% \]
Thus, Assertion A is **true**.

### Step 3: Analyze Reason R
The reason states that the terminal velocity of the spherical body falling through the liquid is inversely proportional to its radius.

This statement is incorrect. As we derived above, the terminal velocity is proportional to the square of the radius, not inversely proportional. Therefore, Reason R is **false**.

### Conclusion
- Assertion A is true.
- Reason R is false.

Thus, the correct answer is that Assertion A is true, and Reason R is false.

### Final Answer
The correct option is **3**: Assertion A is true, and Reason R is false.

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