Dimensions and dimensional analysis — practice questions

Question 1 · difficulty L2 · understanding

$$\text { Match List - I with List - II. }$$ $$\text { List - I }$$ $$\text { List - II }$$ A. Boltzmann constant I. $$\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]$$ B. Stefan's constant II. $$\left[\mathrm{M} \mathrm{~L}^2 \mathrm{~T}^{-1}\right]$$ C. Planck's constant III. $$\left[\mathrm{ML}^2 \mathrm{~T}^{-2} \mathrm{~K}^{-1}\right]$$ D. Gravitational constant IV. $$\left[\mathrm{M} \mathrm{~L}^0 \mathrm{~T}^{-3} \mathrm{~K}^{-4}\right]$$ Choose the correct answer from the options given below :

• A. A-I, B-II, C-III, D-IV
• B. A-IV, B-III, C-II, D-I
• C. A-III, B-IV, C-II, D-I
• D. A-II, B-I, C-IV, D-III

Question 2 · difficulty L2 · understanding

The potential energy of a particle changes with distance $x$ from a fixed origin as $V=\frac{A \sqrt{x}}{x+B}$, where $A$ and $B$ are constant with appropriate dimensions. The dimensions of $A B$ are $\_\_\_\_$

• A. $$\left[\mathrm{M}^1 \mathrm{~L}^{5 / 2} \mathrm{~T}^{-2}\right]$$
• B. $$\left[\mathrm{M}^{3 / 2} \mathrm{~L}^{5 / 2} \mathrm{~T}^{-2}\right]$$
• C. $$\left[\mathrm{M}^1 \mathrm{~L}^2 \mathrm{~T}^{-2}\right]$$
• D. $$\left[\mathrm{M}^1 \mathrm{~L}^{7 / 2} \mathrm{~T}^{-2}\right]$$

Question 3 · difficulty L2 · understanding

$$\text { Match the LIST-I with LIST-II }$$ List - I List - II A. Planck's constant I. $$\mathrm{ML}^2 \mathrm{~T}^{-2}$$ B. Stopping potential II. $$\mathrm{T}^{-1}$$ C. Work function III. $$\mathrm{ML}^2 \mathrm{~T}^{-1}$$ D. Threshold frequency IV. $$\mathrm{ML}^2 \mathrm{~T}^{-3} \mathrm{~A}^{-1}$$ Choose the correct answer from the options given below:

• A. A-III, B-IV, C-I, D-II
• B. A-I, B-II, C-III, D-IV
• C. A-IV, B-III, C-I, D-II
• D. A-I, B-IV, C-III, D-II

Question 4 · difficulty L2 · understanding

The dimensional formula of $\frac{1}{2} \epsilon_0 E^2$ ($\epsilon_0$ = permittivity of vacuum and $E$ = electric field) is $M^a L^b T^c$. The value of $2a - b + c =$ ________.

• A. 0
• B. 1
• C. -1
• D. 2

Question 5 · difficulty L2 · understanding

Match List - I with List - II . List – I List – II A. Coefficient of viscosity B. Surface tension C. Pressure D. Surface energy I. [ML −1 T −2 ] II. [ML 2 T −2 ] III. [ML 0 T −2 ] IV. [ML −1 T −1 ] Choose the correct answer from the options given below :

• A. A-I, B-II, C-IV, D-III
• B. A-IV, B-III, C-I, D-II
• C. A-IV, B-I, C-II, D-III
• D. A-I, B-III, C-II, D-IV

Question 6 · difficulty L2 · understanding

$$\text { Match the LIST-I with LIST-II }$$ List-I List-II A. Spring constant I. M L 2 T − 2 K − 1 M L 2 T − 2 K − 1 ML^(2)T^(-2)K^(-1) B. Thermal conductivity II. ML 0 T − 2 ML 0 T − 2 ML^(0)T^(-2) C. Boltzmann constant III. M L 2 T − 3 A − 2 M L 2 T − 3 A − 2 ML^(2)T^(-3)A^(-2) D. Inductive reactance IV. M L T − 3 K − 1 M L T − 3 K − 1 MLT^(-3)K^(-1) Choose the correct answer from the options given below:

• A. A-III, B-II, C-IV, D-I
• B. A-I, B-IV, C-II, D-III
• C. A-II, B-I, C-IV, D-III
• D. A-II, B-IV, C-I, D-III

Question 7 · difficulty L2 · understanding

Consider a modified Bernoulli equation. $$\left(\mathrm{P}+\frac{A}{B t^2}\right)+\rho g(h+B t)+\frac{1}{2} \rho V^2=\text { constant }$$ If $t$ has the dimension of time then the dimensions of $A$ and $B$ are $\_\_\_\_$ , $\_\_\_\_$ respectively.

• A. $\left[\mathrm{ML}^0 \mathrm{~T}^{-1}\right]$ and $\left[\mathrm{M}^0 \mathrm{LT}\right]$
• B. $\left[\mathrm{ML}^0 \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{M}^0 \mathrm{LT}^{-1}\right]$
• C. $\left[\mathrm{ML}^0 \mathrm{~T}^{-2}\right]$ and $\left[\mathrm{M}^0 \mathrm{LT}^{-2}\right]$
• D. $\left[\mathrm{ML}^0 \mathrm{~T}^{-1}\right]$ and $\left[\mathrm{M}^0 \mathrm{LT}^{-1}\right]$

Question 8 · difficulty L2 · understanding

The dimension of $\sqrt{\frac{\mu_0}{\epsilon_0}}$ is equal to that of: ($\mu_0$ = Vacuum permeability and $\epsilon_0$ = Vacuum permittivity)

• A. Voltage
• B. Inductance
• C. Resistance
• D. Capacitance

Question 9 · difficulty L2 · understanding

Match List - I with List - II . List - I List - II (A) Mass density (I) [ML 2 T −3 ] (B) Impulse (II) [MLT −1 ] (C) Power (III) [ML 2 T 0 ] (D) Moment of inertia (IV) [ML −3 T 0 ] Choose the correct answer from the options given below :

• A. (A)-(IV), (B)-(II), (C)-(III), (D)-(I)
• B. (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
• C. (A)-(IV), (B)-(III), (C)-(I), (D)-(II)
• D. (A)-(IV), (B)-(II), (C)-(I), (D)-(III)

Question 10 · difficulty L2 · understanding

If $\mu_0$ and $\epsilon_0$ are the permeability and permittivity of free space, respectively, then the dimension of $\left(\frac{1}{\mu_0 \epsilon_0}\right)$ is :

• A. $\mathrm{T}^2 / \mathrm{L}$
• B. $\mathrm{L}^2 / \mathrm{T}^2$
• C. $\mathrm{T}^2 / \mathrm{L}^2$
• D. $\mathrm{L} / \mathrm{T}^2$