Dimensional Homogeneity
In a physics tutorial a relation v = a + b t + c t^2 is given for the speed of a moving trolley at time t. What are the dimensions of the constant c in this equation?
Select the correct option:
Solution
[M0L1T−3]
The principle of dimensional homogeneity, presented in NCERT Class 11, Chapter 2 (Units and Measurements), requires every additive term in a valid physical equation to share the same dimensions. Here the left side v is a speed with dimensions [L^{1}T^{-1}], so each term on the right must also have dimensions of speed. For the term c t^2 to equal a speed, c must satisfy [c][T^{2}] = [L^{1}T^{-1}]. Solving gives [c] = [L^{1}T^{-1}] / [T^{2}] = [M^{0}L^{1}T^{-3}]. The option [M^{0}L^{1}T^{-1}] is wrong because it equals the dimensions of b t, not c. The option [M^{0}L^{1}T^{-2}] is wrong as it matches acceleration, which would suit b not c. The option [M^{0}L^{0}T^{-1}] is wrong since it omits the length dimension entirely. A final check confirms that multiplying [L^{1}T^{-3}] by [T^{2}] indeed restores the speed dimensions, validating the result.
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About This Question
- Subject
- physics
- Chapter
- physics and measurement
- Topic
- dimensional homogeneity
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
[M0L1T−3]
The principle of dimensional homogeneity, presented in NCERT Class 11, Chapter 2 (Units and Measurements), requires every additive term in a valid physical equation to share the same dimensions. Here the left side v is a speed with dimensions [L^{1}T^{-1}], so each term on the right must also have dimensions of speed. For the term c t^2 to equal a speed, c must satisfy [c][T^{2}] = [L^{1}T^{-1}]. Solving gives [c] = [L^{1}T^{-1}] / [T^{2}] = [M^{0}L^{1}T^{-3}]. The option [M^{0}L^{1}T^{-1}] is wrong because it equals the dimensions of b t, not c. The option [M^{0}L^{1}T^{-2}] is wrong as it matches acceleration, which would suit b not c. The option [M^{0}L^{0}T^{-1}] is wrong since it omits the length dimension entirely. A final check confirms that multiplying [L^{1}T^{-3}] by [T^{2}] indeed restores the speed dimensions, validating the result.
This medium difficulty physics question is from the chapter physics and measurement, covering the topic of dimensional homogeneity. It appeared in the 2025 exam.
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