Question 60: Properties of Matter

Question

Given below are two statements: one is labelled as Assertion $$\mathbf{A}$$ and the other is labelled as Reason $$\mathbf{R}$$
Assertion A : A spherical body of radius $$(5 \pm 0.1) \mathrm{mm}$$ 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.
In the light of the above statements, choose the correct answer from the options given below

Accepted Answer

The terminal velocity $$v_t$$ of a spherical body falling through a viscous fluid is given by Stokes' Law, which states that:
$$ v_t = \frac{2}{9}\frac{(ho_s - ho_f)gr^2}{\eta} $$
where:
$ ho_s $ is the density of the sphere
$ ho_f $ is the density of the fluid
$ g $ is the acceleration due to gravity
$ r $ is the radius of the sphere
$ \eta $ is the dynamic viscosity of the fluid
As per Stokes' Law, the terminal velocity is proportional to the square of the radius of the sphere (since the radius term $$r^2$$ is in the numerator).
Note that Reason R states that the terminal velocity is "inversely proportional" to its radius, which is contrary to the relationship presented by Stokes' Law. Therefore, Reason R is false.
Moving on to Assertion A, we can consider the percentage error in the radius to determine the percentage error in the terminal velocity. If the radius $$r$$ has an error of $$ \pm 0.1 \mathrm{mm} $$ at $$ 5 \mathrm{mm} $$, then the relative error in the radius is:
$$ \frac{0.1}{5} = 0.02 	ext{ or } 2\% $$
Since the terminal velocity varies with the square of the radius, the percentage error in the terminal velocity would be twice the percentage error in the radius.
$$ 	ext{Percentage error in } v_t = 2 	imes 	ext{ (Percentage error in } r) $$
$$ 	ext{Percentage error in } v_t = 2 	imes 2\% = 4\% $$
This is in agreement with Assertion A, making it true.
Given this analysis, the correct statement is:
Option B: A is true but $$\mathbf{R}$$ is false.

Answer

A is false but $$\mathbf{R}$$ is true

Answer

$$\mathrm{A}$$ is true but $$\mathbf{R}$$ is false

Answer

Both $$\mathbf{A}$$ and $$\mathbf{R}$$ are true but $$\mathbf{R}$$ is NOT the correct explanation of $$\mathbf{A}$$

Answer

Both $$\mathbf{A}$$ and $$\mathbf{R}$$ are true and $$\mathbf{R}$$ is the correct explanation of $$\mathbf{A}$$

Properties of Matter

Choose the correct answer: