Dimensional Analysis Examples

Mathematical Modeling · Axiom Academy

EXAMPLE Dimensional Analysis Examples Worked examples applying the Buckingham Pi Theorem to physical problems A simple pendulum consists of a mass m suspended from a pivot point by a massless string of length L . When displaced from equilibrium by a small angle and released, it oscillates with a period T . We want to determine how the period depends on the physical parameters: length L , gravitational acceleration g , mass m , and initial angle θ₀ . Variables and Their Dimensions We have n = 5 variables: T, L, g, m, θ₀ The base dimensions involved are k = 3 : Mass [M], Length [L], Time [T] By the Buckingham Pi Theorem, we expect n - k = 2 dimensionless groups. We select L , g , and m as repeating variables because: They include all three base dimensions (M, L, T) They are independent (no one can be formed from the others) They do not include the quantity we're solving for (T) Setting the exponents to make dimensions vanish: Second Pi group : Since θ₀ is already dimensionless, Key Insight: The mass m does not appear in the result! Dimensional analysis reveals that the period of a simple pendulum is independent of mass. The period scales as the square root of L/g. For small angles, the function f(θ₀) is approximately constant, giving the familiar formula . Why doesn't the mass appear in the final expression for the pendulum period?

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