Dimensional Analysis Questions

---

Very Short Answers Questions [1 Mark Questions]

Ques. Define dimensional analysis.

Ans. Dimensional analysis is the study of the connections between different physical quantities in engineering and science by determining their base quantities and units of measurement and tracking these dimensions as calculations or comparisons are performed.

Ques. Define the dimensions of a physical quantity.

Ans. The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to present that quantity.

Ques. State the principle of homogeneity of dimensions.

Ans. According to the principle of homogeneity, the dimensions of each term of a dimensional equation on both sides are the same.

Ques. Identify the dimensionless quantity.

1. Strain
2. Angle
3. Specific gravity
4. All of the above

Ans. The correct answer is d. All of the above

Explanation: A dimensionless quantity is one that has no physical dimension. All of the mentioned options are dimensionless quantities.

Ques. Identify the dimensional constant.

1. Momentum
2. Planck’s constant
3. Specific gravity
4. Force

Ans. The correct answer is b. Planck’s constant

Explanation: A dimensional constant is a fixed-valued physical quantity with dimensions. Among the given options Planck’s constant is a dimensional constant quantity.

---

Short Answers Questions [2 Marks Questions]

Ques. Define the dimensional formula with examples.

Ans. The expression of a physical quantity in terms of its dimensions is called a dimensional formula. 

For example:

The dimensional formula of force is [MLT^-2]

The dimension formula of acceleration is [LT^-2]

The dimensional formula of density is [ML^-3]

Ques. What is a dimensional equation?

Ans. An equation that contains a physical quantity on one side and its dimensional formula on the other side, is called the dimensional equation of that quantity.

For example:

Speed, [v] = [M⁰LT^-1]

Area, [A] = [M⁰L²T⁰]

Ques. What are dimensional variables?

Ans. Physical quantities with a numerical value and a specific dimension are referred to as dimensional variables. For example: velocity, Length, Acceleration, etc.

Ques. Define dimensionless quantities.

Ans. A dimensionless quantity, also known as a bare quantity, pure quantity, or quantity of dimension one is one that has no physical dimension.

Read More: 

---

Long Answers Questions [3 Marks Questions]

Ques. What are the applications of dimensional analysis?

Ans. The following are the applications of dimensional analysis

• The correctness of an equation or any other physical relationship may be tested using the Homogeneity principle.
• The relationship is correct if the dimensions of the left and right sides are the same.
• It is used to convert one physical quantity unit to another.
• It is also used to demonstrate the physical quantity's nature.
• Dimensional expressions can be expressed algebraically.
• The most common use of dimensional analysis is to derive formulae.

Ques. What are the limitations of Dimensional Analysis?

Ans. The following are the limitations of Dimensional Analysis

• The dimensionless constant value cannot be calculated with this method.
• This method cannot analyze equations with trigonometric, exponential, or logarithmic terms.
• If a physical quantity in mechanics depends on more than three factors, a relationship between them cannot be created since equalizing the powers of M, L, and T yields only three equations.
• There is no indication of whether the quantity is vector or scalar.

Ques. The energy E of a photon of light is related to its frequency f as E = hf. Here, h is Planck’s constant. The value of Planck’s constant is h = 6.62 x 10^-34 J-s. Covert it into the CGS system of units.

Ans. First, we will find the dimensional formula of Planck’s constant with the help of the given formula of energy of a photon.

E = hf

⇒ h = E/f

⇒ [h] = [ML²T^-2]/[T^-1]

⇒ [h] = [ML²T^-1]

Given, in SI unit, the numerical value of Planck’s constant is, n₁= 6.62 x 10^-34 

The numerical value of Planck’s constant in the CGS unit can be given by

n₂ = n₁ [(M₁/M₂)¹ (L₁/L₂)² (T₁/T₂)^-1]

⇒ n₂ = 6.62 x 10^-34 [(1 kg/1g)¹ (1 m/1 cm)² (1 s/1 s)^-1]

⇒ n₂ = 6.62 x 10^-34 x 10³ x 10⁴ x 1

⇒ n₂ = 6.62 x 10^-27

The value of Planck’s constant in the CGS unit system is 6.62 x 10^-27 erg-s.

---

Very Long Answers Questions [5 Marks Questions]

Ques. Find the dimensions of the following physical quantities

1. Frictional force
2. Charge
3. Refraction index

Ans. All the given physical quantities are derived quantities. First, we will see which fundamental quantities are containing the given derived quantities. Then we write their dimensions.

1. Frictional force is a type of force that opposes the motion of an object. It contains the dimensions of force. i.e. 

[Frictional force] = [Force]

[Force] = [mass] x [acceleration]

⇒ [Force] = [mass] x [velocity]/[time]

⇒ [Force] = [mass] x [displacement]/[time] x [time]

Now we will the dimensions of each fundamental quantities

[Frictional Force] = [M][L]/[T][T]

⇒ [Frictional Force] = [M L T^-2]

2. The formula of charge can be given by

[Charge] = [Current][Time]

[Charge] = [A][T]

⇒ [Charge] = [AT]

100. The ratio of the speed of light in a vacuum to the speed of light in a particular medium is called the refractive index of the medium.

[Refractive index] = [Speed of light in vacuum]/[Speed of light in medium]

⇒ [Refractive index] = [L T^-1]/[L T^-1]

⇒ [Refractive index] = [L⁰ T⁰]

Hence the refractive index is a dimensionless quantity.

Ques. How do you convert 1 Newton to dyne by dimensional formula?

Ans. The dimensional formula of force is given by

[Force] = [M L T^-2]

Newton is the SI unit of force. 1 Newton is equal to 1 kg m s^-2

Given the numerical value of force in SI unit is n₁ = 1

Dyne is the CGS unit of force. 1 dyne is equal to 1 g cm s^-2

The numerical value of force in the CGS unit can be given by

n₂ = n₁ [(M₁/M₂)¹ (L₁/L₂)¹ (T₁/T₂)^-2]

⇒ n₂ = 1 x [(1 kg/1g)¹ (1 m/1 cm)¹ (1 s/1 s)^-2]

⇒ n₂ = 1 x 10³ x 10² x 1^-2

⇒ n₂ = 10⁵

Therefore, 1 Newton = 10⁵ dyne

Ques. How do you convert 1 joule to erg by dimensional formula?

Ans. The dimensional formula of energy is given by

[Force] = [M L² T^-2]

Joule is the SI unit of energy. 1 joule is equal to 1 kg m² s^-2

Given the numerical value of energy in SI unit is n₁ = 1

Erg is the CGS unit of energy. 1 erg is equal to 1 g cm² s^-2

The numerical value of energy in the CGS unit can be given by

n₂ = n₁ [(M₁/M₂)¹ (L₁/L₂)² (T₁/T₂)^-2]

⇒ n₂ = 1 x [(1 kg/1g)¹ (1 m/1 cm)² (1 s/1 s)^-2]

⇒ n₂ = 1 x 10³ x 10⁴ x 1^-2

⇒ n₂ = 10⁷

Therefore, 1 joule = 10⁷ erg

---

Previous Year Questions

1. Extraction of metal from the ore cassiterite involves...[JEE Advanced 2011]
2. Commonly used vectors for human genome sequencing are...[NEET UG 2014]
3. Interfascicular cambium and cork cambium are formed due to​..
4. Pneumotaxic centre is present in​...[UP CPMT 2007]
5. Reaction of HBr with propene in the presence of peroxide gives….[NEET UG 2004]
6. Assuming the expression for the pressure exerted by the gas on the walls of the container, it can be shown that pressure is...[MHT CET 2016]
7. Which among the following is the strongest acid?...[TS EAMCET 2017]
8. Isopropyl alcohol on oxidation forms​..
9. A vector is not changed if​..
10. Which of the following arrangements does not represent the correct order of the property stated against it?...[JEE Main 2013]
11. The major product of the following reaction is​...[JEE Main 2019]
12. Major product of the following reaction is..[JEE Main 2023]
13. The percentage of nitrogen in urea is about..
14. The electric field at a point is​
15. Which of the following statements is true?​..[JKCET 2006]

For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates

---

Check-Out: