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The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair (s)
- a)Torque and Work
- b)Angular momentum and Work
- c)Energy and Young’s modulus
- d)Light year and Wavelength
Correct answer is option 'A,D'. Can you explain this answer?
Verified Answer
The dimensions of the quantities in one (or more) of the following pai...
t = F × r × sin θ; W = F × d × cosqθ
Dimensionally, light year = wavelength = [L]
Dimensionally, light year = wavelength = [L]
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The dimensions of the quantities in one (or more) of the following pai...
Understanding the Dimensions of Physical Quantities
To analyze the pairs of quantities and identify those with the same dimensions, we need to understand their definitions and the dimensions associated with them.
1. Torque and Work
- Torque (τ): Defined as the rotational equivalent of linear force, its formula is τ = r × F, where:
- \( r \) (distance) has dimensions of \( [L] \) (length).
- \( F \) (force) has dimensions of \( [M][L][T^{-2}] \) (mass, length, time).
Thus, the dimensions of Torque are:
\[ [τ] = [L][M][L][T^{-2}] = [M][L^{2}][T^{-2}] \]
- Work (W): Defined as the energy transferred when a force is applied over a distance, its formula is W = F × d. Thus, the dimensions of Work are also:
\[ [W] = [M][L][T^{-2}][L] = [M][L^{2}][T^{-2}] \]
- Conclusion: Since both Torque and Work have the same dimensions, they are a matching pair.
2. Angular Momentum and Work
- Angular Momentum (L): Defined as L = r × p, where:
- \( p \) (linear momentum) has dimensions of \( [M][L][T^{-1}] \).
Thus, the dimensions of Angular Momentum are:
\[ [L] = [L][M][L][T^{-1}] = [M][L^{2}][T^{-1}] \]
- Conclusion: Angular Momentum has different dimensions compared to Work.
3. Energy and Young’s Modulus
- Energy: Has dimensions of \( [M][L^{2}][T^{-2}] \).
- Young’s Modulus: Defined as stress/strain, its dimensions are given by:
- Stress has dimensions of \( [M][L^{-1}][T^{-2}] \) and strain is dimensionless.
Thus, the dimensions of Young's Modulus are:
\[ [Y] = [M][L^{-1}][T^{-2}] \]
- Conclusion: Energy and Young's Modulus do not share the same dimensions.
4. Light Year and Wavelength
- Light Year: A measure of distance, defined as the distance light travels in one year, with dimensions of \( [L] \) (length).
- Wavelength: Also a measure of distance, with dimensions of \( [L] \).
- Conclusion: Both Light Year and Wavelength share the same dimensions.
Final Conclusion
- The pairs with the same dimensions are Torque and Work (Option A) and Light Year and Wavelength (Option D).
To analyze the pairs of quantities and identify those with the same dimensions, we need to understand their definitions and the dimensions associated with them.
1. Torque and Work
- Torque (τ): Defined as the rotational equivalent of linear force, its formula is τ = r × F, where:
- \( r \) (distance) has dimensions of \( [L] \) (length).
- \( F \) (force) has dimensions of \( [M][L][T^{-2}] \) (mass, length, time).
Thus, the dimensions of Torque are:
\[ [τ] = [L][M][L][T^{-2}] = [M][L^{2}][T^{-2}] \]
- Work (W): Defined as the energy transferred when a force is applied over a distance, its formula is W = F × d. Thus, the dimensions of Work are also:
\[ [W] = [M][L][T^{-2}][L] = [M][L^{2}][T^{-2}] \]
- Conclusion: Since both Torque and Work have the same dimensions, they are a matching pair.
2. Angular Momentum and Work
- Angular Momentum (L): Defined as L = r × p, where:
- \( p \) (linear momentum) has dimensions of \( [M][L][T^{-1}] \).
Thus, the dimensions of Angular Momentum are:
\[ [L] = [L][M][L][T^{-1}] = [M][L^{2}][T^{-1}] \]
- Conclusion: Angular Momentum has different dimensions compared to Work.
3. Energy and Young’s Modulus
- Energy: Has dimensions of \( [M][L^{2}][T^{-2}] \).
- Young’s Modulus: Defined as stress/strain, its dimensions are given by:
- Stress has dimensions of \( [M][L^{-1}][T^{-2}] \) and strain is dimensionless.
Thus, the dimensions of Young's Modulus are:
\[ [Y] = [M][L^{-1}][T^{-2}] \]
- Conclusion: Energy and Young's Modulus do not share the same dimensions.
4. Light Year and Wavelength
- Light Year: A measure of distance, defined as the distance light travels in one year, with dimensions of \( [L] \) (length).
- Wavelength: Also a measure of distance, with dimensions of \( [L] \).
- Conclusion: Both Light Year and Wavelength share the same dimensions.
Final Conclusion
- The pairs with the same dimensions are Torque and Work (Option A) and Light Year and Wavelength (Option D).
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The dimensions of the quantities in one (or more) of the following pairs are the same. Identify the pair (s)a)Torque and Workb)Angular momentum and Workc)Energy and Young’s modulusd)Light year and WavelengthCorrect answer is option 'A,D'. Can you explain this answer?
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