Impulse is defined as the product of the average force applied to an object and the time duration over which that force acts. What is the dimensional formula of impulse in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

Latent heat $L$ is defined as the thermal energy transferred per unit mass during a phase change at constant temperature ($L = Q / m$). What is the dimensional formula of latent heat in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

The speed $v$ of an object at time $t$ is given by $v = v_0 e^{-k t}$, where $v_0$ is the initial speed. What is the dimensional formula of the constant $k$ in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

Which of the following statements identifies a fundamental limitation of dimensional analysis?

A) It cannot be used to check whether an algebraic physical equation is dimensionally consistent.

B) It cannot determine the numerical value of dimensionless proportionality constants in derived relationships.

C) It cannot be used to find the base dimensional formula of derived physical quantities.

D) It cannot be applied to convert physical quantities from one system of units to another.

According to the principle of dimensional homogeneity, which of the following equations is dimensionally inconsistent, where $x$ represents displacement, $v$ represents speed, $a$ represents acceleration, and $t$ represents time?

B) $x = v t + \frac{1}{2} a t^2$

D) $x = a t + \frac{1}{2} v t^2$

Power is defined as the rate at which work is done per unit time. What is the dimensional formula of power in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

Wave intensity is defined as the radiant power transmitted per unit area perpendicular to the direction of propagation. What is the dimensional formula of wave intensity in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

In the van der Waals equation of state, $\left(P + \frac{a}{V^2}\right)(V - b) = R T$, $P$ represents pressure and $V$ represents volume. Based on the principle of dimensional homogeneity, what is the dimensional formula of the constant $a$ in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

Surface tension is defined as the force acting per unit length along a line in the surface of a liquid. What is the dimensional formula of surface tension in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

The coefficient of dynamic viscosity $\eta$ of a fluid is defined as the ratio of shear stress to the velocity gradient. What is the dimensional formula of $\eta$ in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

Energy density is defined as the amount of energy stored per unit volume of a physical system. What is the dimensional formula of energy density in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

The pressure gradient in a fluid flow is defined as the rate of change of pressure with respect to distance along a given direction. What is the dimensional formula of the pressure gradient in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

The speed $v$ of a transverse wave on a stretched string depends on the tension force $F$ and the linear mass density $\mu$ (mass per unit length) according to the relation $v = k F^x \mu^y$, where $k$ is a dimensionless constant. Using dimensional analysis, what are the values of the exponents $x$ and $y$?

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

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

In Newton's law of universal gravitation, the attractive force between two point masses is given by $F = G \frac{m_1 m_2}{r^2}$, where $F$ is force, $m_1$ and $m_2$ are masses, and $r$ is the separation distance. What is the dimensional formula of the universal gravitational constant $G$ in terms of base dimensions $\text{M}$, $\text{L}$, and $\text{T}$?

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

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

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

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

In the hydrodynamic drag equation $F = k v^2$, $F$ represents force and $v$ represents speed. What is the dimension of the constant $k$ in terms of base dimensions mass (M), length (L), and time (T)?

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

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

Dimensional analysis Knowledge-Based Download PDF

Physics · Introduction to Physics · Physics · Measurement & Units

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

Questions

1.
Impulse is defined as the product of the average force applied to an object and the time duration over which that force acts. What is the dimensional formula of impulse in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M} \text{L} \text{T}^{-2}$
- B) $\text{M} \text{L}^2 \text{T}^{-1}$
- C) $\text{M} \text{L} \text{T}^{-1}$
- D) $\text{M} \text{L}^2 \text{T}^{-2}$
2.
Latent heat $L$ is defined as the thermal energy transferred per unit mass during a phase change at constant temperature ( $L = Q / m$ ). What is the dimensional formula of latent heat in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M} \text{L}^2 \text{T}^{-2}$
- B) $\text{M}^0 \text{L}^2 \text{T}^{-2}$
- C) $\text{M}^0 \text{L} \text{T}^{-2}$
- D) $\text{M}^{-1} \text{L}^2 \text{T}^{-2}$
3.
The speed $v$ of an object at time $t$ is given by $v = v_0 e^{-k t}$ , where $v_0$ is the initial speed. What is the dimensional formula of the constant $k$ in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M}^0 \text{L}^0 \text{T}$
- B) $\text{M}^0 \text{L}^0 \text{T}^{-1}$
- C) $\text{M}^0 \text{L} \text{T}^{-1}$
- D) $\text{M}^0 \text{L}^0 \text{T}^{-2}$
4.
Which of the following statements identifies a fundamental limitation of dimensional analysis?
- A) It cannot be used to check whether an algebraic physical equation is dimensionally consistent.
- B) It cannot determine the numerical value of dimensionless proportionality constants in derived relationships.
- C) It cannot be used to find the base dimensional formula of derived physical quantities.
- D) It cannot be applied to convert physical quantities from one system of units to another.
5.
According to the principle of dimensional homogeneity, which of the following equations is dimensionally inconsistent, where $x$ represents displacement, $v$ represents speed, $a$ represents acceleration, and $t$ represents time?
- A) $v^2 = 2 a x$
- B) $x = v t + \frac{1}{2} a t^2$
- C) $a = \frac{v^2}{x}$
- D) $x = a t + \frac{1}{2} v t^2$
6.
Which of the following physical quantities is dimensionless, having the dimensional formula $\text{M}^0 \text{L}^0 \text{T}^0$ ?
- A) Relative density
- B) Mass density
- C) Specific volume
- D) Linear mass density
7.
Power is defined as the rate at which work is done per unit time. What is the dimensional formula of power in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M}\text{L}^2\text{T}^{-3}$
- B) $\text{M}\text{L}^2\text{T}^{-2}$
- C) $\text{M}\text{L}\text{T}^{-3}$
- D) $\text{M}\text{L}^2\text{T}^{-1}$
8.
Wave intensity is defined as the radiant power transmitted per unit area perpendicular to the direction of propagation. What is the dimensional formula of wave intensity in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M} \text{L}^2 \text{T}^{-2}$
- B) $\text{M} \text{L}^2 \text{T}^{-3}$
- C) $\text{M} \text{L}^0 \text{T}^{-3}$
- D) $\text{M} \text{L}^{-1} \text{T}^{-2}$
9.
In the van der Waals equation of state, $\left(P + \frac{a}{V^2}\right)(V - b) = R T$ , $P$ represents pressure and $V$ represents volume. Based on the principle of dimensional homogeneity, what is the dimensional formula of the constant $a$ in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M}\text{L}^5\text{T}^{-2}$
- B) $\text{M}\text{L}^{-1}\text{T}^{-2}$
- C) $\text{M}\text{L}^2\text{T}^{-2}$
- D) $\text{M}\text{L}^4\text{T}^{-2}$
10.
Surface tension is defined as the force acting per unit length along a line in the surface of a liquid. What is the dimensional formula of surface tension in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M T}^{-2}$
- B) $\text{M L T}^{-2}$
- C) $\text{M L}^{-1} \text{T}^{-2}$
- D) $\text{M L}^2 \text{T}^{-2}$
11.
The coefficient of dynamic viscosity $\eta$ of a fluid is defined as the ratio of shear stress to the velocity gradient. What is the dimensional formula of $\eta$ in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- 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) $\text{M} \text{L}^{-2} \text{T}^{-1}$
12.
If $P$ represents pressure and $\rho$ represents mass density, which of the following combinations has the dimension of speed?
- A) $\sqrt{\frac{P}{\rho}}$
- B) $\frac{P}{\rho}$
- C) $\sqrt{P \rho}$
- D) $\frac{P^2}{\rho}$
13.
Energy density is defined as the amount of energy stored per unit volume of a physical system. What is the dimensional formula of energy density in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M L}^2 \text{T}^{-2}$
- B) $\text{M L}^5 \text{T}^{-2}$
- C) $\text{M L}^{-1} \text{T}^{-2}$
- D) $\text{M L}^{-2} \text{T}^{-1}$
14.
The pressure gradient in a fluid flow is defined as the rate of change of pressure with respect to distance along a given direction. What is the dimensional formula of the pressure gradient in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $[\text{M}\text{L}^{-1}\text{T}^{-2}]$
- B) $[\text{M}\text{L}^{-2}\text{T}^{-2}]$
- C) $[\text{M}\text{L}\text{T}^{-2}]$
- D) $[\text{M}\text{L}^{-3}\text{T}^{-2}]$
15.
The speed $v$ of a transverse wave on a stretched string depends on the tension force $F$ and the linear mass density $\mu$ (mass per unit length) according to the relation $v = k F^x \mu^y$ , where $k$ is a dimensionless constant. Using dimensional analysis, what are the values of the exponents $x$ and $y$ ?
- A) $x = 1$ and $y = -1$
- B) $x = -\frac{1}{2}$ and $y = \frac{1}{2}$
- C) $x = \frac{1}{2}$ and $y = -\frac{1}{2}$
- D) $x = 2$ and $y = -2$
16.
In Newton's law of universal gravitation, the attractive force between two point masses is given by $F = G \frac{m_1 m_2}{r^2}$ , where $F$ is force, $m_1$ and $m_2$ are masses, and $r$ is the separation distance. What is the dimensional formula of the universal gravitational constant $G$ in terms of base dimensions $\text{M}$ , $\text{L}$ , and $\text{T}$ ?
- A) $\text{M} \text{L}^2 \text{T}^{-2}$
- B) $\text{M}^{-1} \text{L}^2 \text{T}^{-1}$
- C) $\text{M}^2 \text{L}^{-3} \text{T}^2$
- D) $\text{M}^{-1} \text{L}^3 \text{T}^{-2}$
17.
In the hydrodynamic drag equation $F = k v^2$ , $F$ represents force and $v$ represents speed. What is the dimension of the constant $k$ in terms of base dimensions mass (M), length (L), and time (T)?
- A) $\text{M L T}^{-1}$
- B) $\text{M L}^{-1}$
- C) $\text{M L}^{2}\text{T}^{-2}$
- D) $\text{M L}^{-2}\text{T}^{-1}$

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