According to the principle of dimensional homogeneity, which condition must be satisfied by all terms added or subtracted in a physically valid equation?

A) They must contain the exact same set of physical variables.

B) They must have identical dimensions in terms of the fundamental quantities.

C) They must have equal numerical magnitudes when evaluated in SI units.

D) They must all be dimensionless scalar quantities.

The volumetric flow rate $Q$ of a fluid is defined as the volume of fluid passing through a cross-sectional area per unit time. What are the dimensions of volumetric flow rate in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}^0\text{L}^2\text{T}^{-1}$

B) $\text{M}^1\text{L}^3\text{T}^{-1}$

C) $\text{M}^0\text{L}^3\text{T}^{-2}$

D) $\text{M}^0\text{L}^3\text{T}^{-1}$

The intensity of a wave is defined as the energy transmitted per unit time across a unit area. What are the dimensions of wave intensity in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

B) $\text{M}\text{L}^2\text{T}^{-3}$

C) $\text{M}\text{L}^{-1}\text{T}^{-2}$

D) $\text{M}\text{L}\text{T}^{-3}$

Energy density is defined as the amount of energy stored per unit volume. What are the dimensions of energy density in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M} \text{L}^{2} \text{T}^{-2}$

B) $\text{M} \text{L} \text{T}^{-2}$

C) $\text{M} \text{L}^{-2} \text{T}^{-2}$

D) $\text{M} \text{L}^{-1} \text{T}^{-2}$

Impulse is defined as the product of the average force acting on an object and the time interval over which it acts. What are the dimensions of impulse in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M} \text{L} \text{T}^{-2}$

B) $\text{M} \text{L} \text{T}^{-1}$

C) $\text{M} \text{L}^2 \text{T}^{-1}$

D) $\text{M} \text{L}^2 \text{T}^{-2}$

The pressure gradient along a pipe is defined as the change in pressure per unit length. What are the dimensions of the pressure gradient in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}\text{L}^{-2}\text{T}^{-2}$

B) $\text{M}\text{L}^{-1}\text{T}^{-2}$

C) $\text{M}\text{L}\text{T}^{-2}$

D) $\text{M}\text{L}^{-3}\text{T}^{-2}$

Power is defined as the rate at which work is done over time. What are the dimensions of power in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}\text{L}\text{T}^{-2}$

B) $\text{M}\text{L}^2\text{T}^{-3}$

C) $\text{M}\text{L}^2\text{T}^{-2}$

D) $\text{M}\text{L}^{-1}\text{T}^{-2}$

Surface tension is defined as the force exerted per unit length along a line on the surface of a liquid. What are the dimensions of surface tension in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M} \text{L} \text{T}^{-2}$

B) $\text{M} \text{L}^{-1} \text{T}^{-2}$

C) $\text{M} \text{L}^2 \text{T}^{-2}$

In the van der Waals equation of state, $\left(P + \frac{a}{V^2}\right)(V - b) = nRT$, where $P$ represents pressure and $V$ represents volume, what are the dimensions of the constant $a$ in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}\text{L}^2\text{T}^{-2}$

B) $\text{M}\text{L}^5\text{T}^{-2}$

C) $\text{M}\text{L}^{-1}\text{T}^{-2}$

D) $\text{M}\text{L}^4\text{T}^{-2}$

Young's modulus of a material is defined as the ratio of tensile stress (force per unit area) to tensile strain (fractional change in length). What are the dimensions of Young's modulus in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}\text{L}^{-1}\text{T}^{-2}$

B) $\text{M}\text{L}\text{T}^{-2}$

C) $\text{M}\text{L}^2\text{T}^{-2}$

D) $\text{M}\text{L}^{-2}\text{T}^{-1}$

According to Stokes' law, the drag force acting on a sphere moving through a fluid is given by $F = 6\pi \eta r v$, where $F$ is force, $r$ is radius, and $v$ is speed. What are the dimensions of the dynamic viscosity $\eta$ in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}\text{L}^{-1}\text{T}^{-1}$

B) $\text{M}\text{L}\text{T}^{-1}$

C) $\text{M}\text{L}^{-1}\text{T}^{-2}$

D) $\text{M}\text{L}^{-2}\text{T}^{-1}$

Planck's constant $h$ relates the energy $E$ of a photon to its frequency $f$ through the equation $E = hf$. What are the dimensions of $h$ in terms of mass (M), length (L), and time (T)?

A) $\text{M}\text{L}\text{T}^{-1}$

B) $\text{M}\text{L}^2\text{T}^{-1}$

C) $\text{M}\text{L}^2\text{T}^{-2}$

D) $\text{M}\text{L}^3\text{T}^{-1}$

Which of the following represents a fundamental limitation of dimensional analysis when deriving physical relationships?

A) It cannot check whether a physical equation is dimensionally consistent.

B) It cannot convert the magnitude of a physical quantity between different unit systems.

C) It cannot deduce the dimensions of an unknown constant in a homogeneous equation.

D) It cannot determine the numerical value of dimensionless proportionality constants.

The velocity $v$ of a particle moving through a viscous medium decreases with time $t$ according to the relation $v(t) = v_0 e^{-k t}$, where $v_0$ is the initial velocity. What are the dimensions of the constant $k$ in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}^0 \text{L}^1 \text{T}^{-1}$

B) $\text{M}^0 \text{L}^0 \text{T}^1$

C) $\text{M}^1 \text{L}^0 \text{T}^{-1}$

D) $\text{M}^0 \text{L}^0 \text{T}^{-1}$

Based on dimensional analysis, which of the following expressions is dimensionally consistent for the speed $v$ of sound in a fluid, where $P$ denotes pressure, $\rho$ denotes density, and $C$ is a dimensionless constant?

A) $v = C \left(\frac{P}{\rho}\right)$

C) $v = C \sqrt{\frac{P}{\rho}}$

D) $v = C \sqrt{\frac{\rho}{P}}$

The period of oscillation $T$ of a simple pendulum depends on its length $l$ and the acceleration due to gravity $g$ according to the relation $T = k l^x g^y$, where $k$ is a dimensionless constant. Based on dimensional analysis, what are the values of the exponents $x$ and $y$?

B) $x = \frac{1}{2}$, $y = \frac{1}{2}$

C) $x = 1$, $y = -\frac{1}{2}$

D) $x = \frac{1}{2}$, $y = -\frac{1}{2}$

Which of the following pairs of physical quantities has identical dimensions?

C) Pressure and surface tension

Which of the following expressions represents the dimensions of the universal gravitational constant, $G$, in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}\text{L}^2\text{T}^{-2}$

B) $\text{M}\text{L}^3\text{T}^{-2}$

C) $\text{M}^{-1}\text{L}^2\text{T}^{-2}$

D) $\text{M}^{-1}\text{L}^3\text{T}^{-2}$

In the equation $F = a t + b t^2$, where $F$ denotes force and $t$ denotes time, which of the following represents the dimensions of the constant $a$ in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$)?

A) $\text{M}\text{L}\text{T}^{-2}$

B) $\text{M}\text{L}\text{T}^{-3}$

C) $\text{M}\text{L}^2\text{T}^{-3}$

D) $\text{M}\text{L}\text{T}^{-1}$

Dimensional analysis Knowledge-Based Practice

Physics · Introduction to Physics · Physics · Measurement & Units

20 Physics practice questions on Dimensional analysis. Questions that assess core concepts and understanding.

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Questions

1.
According to the principle of dimensional homogeneity, which condition must be satisfied by all terms added or subtracted in a physically valid equation?
- A) They must contain the exact same set of physical variables.
- B) They must have identical dimensions in terms of the fundamental quantities.
- C) They must have equal numerical magnitudes when evaluated in SI units.
- D) They must all be dimensionless scalar quantities.
2.
The volumetric flow rate $Q$ of a fluid is defined as the volume of fluid passing through a cross-sectional area per unit time. What are the dimensions of volumetric flow rate in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}^0\text{L}^2\text{T}^{-1}$
- B) $\text{M}^1\text{L}^3\text{T}^{-1}$
- C) $\text{M}^0\text{L}^3\text{T}^{-2}$
- D) $\text{M}^0\text{L}^3\text{T}^{-1}$
3.
If force ( $F$ ), velocity ( $v$ ), and time ( $t$ ) are chosen as the fundamental dimensions, what is the dimensional expression for mass?
- A) $F v t^{-1}$
- B) $F v^{-2} t$
- C) $F v^{-1} t$
- D) $F v t^2$
4.
The intensity of a wave is defined as the energy transmitted per unit time across a unit area. What are the dimensions of wave intensity in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}\text{T}^{-3}$
- B) $\text{M}\text{L}^2\text{T}^{-3}$
- C) $\text{M}\text{L}^{-1}\text{T}^{-2}$
- D) $\text{M}\text{L}\text{T}^{-3}$
5.
Energy density is defined as the amount of energy stored per unit volume. What are the dimensions of energy density in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M} \text{L}^{2} \text{T}^{-2}$
- B) $\text{M} \text{L} \text{T}^{-2}$
- C) $\text{M} \text{L}^{-2} \text{T}^{-2}$
- D) $\text{M} \text{L}^{-1} \text{T}^{-2}$
6.
Impulse is defined as the product of the average force acting on an object and the time interval over which it acts. What are the dimensions of impulse in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M} \text{L} \text{T}^{-2}$
- B) $\text{M} \text{L} \text{T}^{-1}$
- C) $\text{M} \text{L}^2 \text{T}^{-1}$
- D) $\text{M} \text{L}^2 \text{T}^{-2}$
7.
The pressure gradient along a pipe is defined as the change in pressure per unit length. What are the dimensions of the pressure gradient in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}\text{L}^{-2}\text{T}^{-2}$
- B) $\text{M}\text{L}^{-1}\text{T}^{-2}$
- C) $\text{M}\text{L}\text{T}^{-2}$
- D) $\text{M}\text{L}^{-3}\text{T}^{-2}$
8.
Power is defined as the rate at which work is done over time. What are the dimensions of power in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}\text{L}\text{T}^{-2}$
- B) $\text{M}\text{L}^2\text{T}^{-3}$
- C) $\text{M}\text{L}^2\text{T}^{-2}$
- D) $\text{M}\text{L}^{-1}\text{T}^{-2}$
9.
Surface tension is defined as the force exerted per unit length along a line on the surface of a liquid. What are the dimensions of surface tension in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M} \text{L} \text{T}^{-2}$
- B) $\text{M} \text{L}^{-1} \text{T}^{-2}$
- C) $\text{M} \text{L}^2 \text{T}^{-2}$
- D) $\text{M} \text{T}^{-2}$
10.
In the van der Waals equation of state, $\left(P + \frac{a}{V^2}\right)(V - b) = nRT$ , where $P$ represents pressure and $V$ represents volume, what are the dimensions of the constant $a$ in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}\text{L}^2\text{T}^{-2}$
- B) $\text{M}\text{L}^5\text{T}^{-2}$
- C) $\text{M}\text{L}^{-1}\text{T}^{-2}$
- D) $\text{M}\text{L}^4\text{T}^{-2}$
11.
Young's modulus of a material is defined as the ratio of tensile stress (force per unit area) to tensile strain (fractional change in length). What are the dimensions of Young's modulus in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}\text{L}^{-1}\text{T}^{-2}$
- B) $\text{M}\text{L}\text{T}^{-2}$
- C) $\text{M}\text{L}^2\text{T}^{-2}$
- D) $\text{M}\text{L}^{-2}\text{T}^{-1}$
12.
According to Stokes' law, the drag force acting on a sphere moving through a fluid is given by $F = 6\pi \eta r v$ , where $F$ is force, $r$ is radius, and $v$ is speed. What are the dimensions of the dynamic viscosity $\eta$ in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}\text{L}^{-1}\text{T}^{-1}$
- B) $\text{M}\text{L}\text{T}^{-1}$
- C) $\text{M}\text{L}^{-1}\text{T}^{-2}$
- D) $\text{M}\text{L}^{-2}\text{T}^{-1}$
13.
Planck's constant $h$ relates the energy $E$ of a photon to its frequency $f$ through the equation $E = hf$ . What are the dimensions of $h$ in terms of mass (M), length (L), and time (T)?
- A) $\text{M}\text{L}\text{T}^{-1}$
- B) $\text{M}\text{L}^2\text{T}^{-1}$
- C) $\text{M}\text{L}^2\text{T}^{-2}$
- D) $\text{M}\text{L}^3\text{T}^{-1}$
14.
Which of the following represents a fundamental limitation of dimensional analysis when deriving physical relationships?
- A) It cannot check whether a physical equation is dimensionally consistent.
- B) It cannot convert the magnitude of a physical quantity between different unit systems.
- C) It cannot deduce the dimensions of an unknown constant in a homogeneous equation.
- D) It cannot determine the numerical value of dimensionless proportionality constants.
15.
The velocity $v$ of a particle moving through a viscous medium decreases with time $t$ according to the relation $v(t) = v_0 e^{-k t}$ , where $v_0$ is the initial velocity. What are the dimensions of the constant $k$ in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}^0 \text{L}^1 \text{T}^{-1}$
- B) $\text{M}^0 \text{L}^0 \text{T}^1$
- C) $\text{M}^1 \text{L}^0 \text{T}^{-1}$
- D) $\text{M}^0 \text{L}^0 \text{T}^{-1}$
16.
Based on dimensional analysis, which of the following expressions is dimensionally consistent for the speed $v$ of sound in a fluid, where $P$ denotes pressure, $\rho$ denotes density, and $C$ is a dimensionless constant?
- A) $v = C \left(\frac{P}{\rho}\right)$
- B) $v = C \sqrt{P \rho}$
- C) $v = C \sqrt{\frac{P}{\rho}}$
- D) $v = C \sqrt{\frac{\rho}{P}}$
17.
The period of oscillation $T$ of a simple pendulum depends on its length $l$ and the acceleration due to gravity $g$ according to the relation $T = k l^x g^y$ , where $k$ is a dimensionless constant. Based on dimensional analysis, what are the values of the exponents $x$ and $y$ ?
- A) $x = 1$ , $y = -1$
- B) $x = \frac{1}{2}$ , $y = \frac{1}{2}$
- C) $x = 1$ , $y = -\frac{1}{2}$
- D) $x = \frac{1}{2}$ , $y = -\frac{1}{2}$
18.
Which of the following pairs of physical quantities has identical dimensions?
- A) Work and torque
- B) Force and impulse
- C) Pressure and surface tension
- D) Power and energy
19.
Which of the following expressions represents the dimensions of the universal gravitational constant, $G$ , in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}\text{L}^2\text{T}^{-2}$
- B) $\text{M}\text{L}^3\text{T}^{-2}$
- C) $\text{M}^{-1}\text{L}^2\text{T}^{-2}$
- D) $\text{M}^{-1}\text{L}^3\text{T}^{-2}$
20.
In the equation $F = a t + b t^2$ , where $F$ denotes force and $t$ denotes time, which of the following represents the dimensions of the constant $a$ in terms of mass ( $\text{M}$ ), length ( $\text{L}$ ), and time ( $\text{T}$ )?
- A) $\text{M}\text{L}\text{T}^{-2}$
- B) $\text{M}\text{L}\text{T}^{-3}$
- C) $\text{M}\text{L}^2\text{T}^{-3}$
- D) $\text{M}\text{L}\text{T}^{-1}$

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