Dimensional Analysis
A researcher derives the time period of a simple pendulum and suspects it depends on length and gravitational acceleration only. Using dimensional analysis, how does the period depend on length L?
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About This Question
- Subject
- physics
- Chapter
- physics and measurement
- Topic
- dimensional analysis
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
ProportionaltoL1/2
Dimensional analysis, as introduced in NCERT Class 11, Chapter 2 (Units and Measurements), lets us deduce the form of a relation by matching dimensions on both sides. Assume the period T depends on length as L^{a} and on gravitational acceleration as g^{b}, so T = k L^{a} g^{b}. The dimensions are [T^{1}] = [L^{1}]^{a} [L^{1}T^{-2}]^{b} = [L^{a+b} T^{-2b}]. Matching the time power gives -2b = 1, so b = -1/2, and matching the length power gives a + b = 0, so a = 1/2. Hence the period is proportional to L^{1/2}. The option L^{1} is wrong because it ignores the constraint from the gravity term. The option L^{-1/2} is wrong as it reverses the required sign of the exponent. The option independent of L is wrong since experiments and the derivation both show a clear length dependence. A final check confirms the familiar pendulum formula T = 2π√(L/g) indeed contains the square root of length.
This hard difficulty physics question is from the chapter physics and measurement, covering the topic of dimensional analysis. It appeared in the 2025 exam.
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