Q.(a)
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Start your 14-day free trial to unlock the full solution →(i) Dimensional analysis has three main uses: verifying equations, deriving formulas, and converting units. (ii) Checking (1/2)mv² = mgh by comparing dimensions on both sides shows they are equal (ML²T^-2 = ML²T^-2), confirming the equation is dimensionally consistent.
(i) Applications of Dimensional Analysis:
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To check the correctness of a physical equation (Principle of Homogeneity of Dimensions): an equation can only be physically correct if the dimensions of every term on both sides are identical; dimensional analysis is a quick way to catch obviously wrong formulas (though it cannot catch errors in pure numbers/dimensionless constants).
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To derive a relationship between physical quantities: if we know (or can guess) which quantities a physical quantity depends on, dimensional analysis (by comparing powers of M, L, T on both sides) can be used to derive the form of the relating equation, up to an unknown dimensionless constant.
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To convert a physical quantity's value from one system of units to another (e.g., from CGS to SI), using the fact that the numerical value × unit must remain the same physical quantity, and dimensions tell us exactly how each unit scales.
(ii) Checking (1/2)mv² = mgh by dimensional analysis:
LHS = (1/2) m v²
Dimensions: [M] × [L T^-1]² = [M] × [L² T^-2] = [M L² T^-2]
(Note: the pure number 1/2 is dimensionless and does not affect the dimensional formula.)
RHS = m g h
Dimensions: [M] × [L T^-2] × [L] = [M L² T^-2]
Comparing:
LHS dimensions = M L² T^-2 …
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