What is the dependence of the period of a pendulum’s oscillation...

Question

What is the dependence of the period of a pendulum’s oscillation on the mass, m, of the pendulum. Why?

Answer 1

The period of the pendulum's swing is independent of the mass of the bob on the pendulum. This is due to the fact that the swing can be modeled using simple harmonic motion, as shown below.

From Newton's second law for torque, we have τ = I·α = mgsinθ · L.

Since I = mL², we can simplify to get that α = gsinθ / L.

For small angles of oscillation, we can use the approximation that sinθ ≈ θ, so we have:

α = (g / L) · θ.

Since for simple harmonic motion, α = ω²θ (or a = ω²x), we have:

ω = sqrt(g / L).

Since period is defined as T = 2π / ω, we have that T = 2π * sqrt(L / g).

You can see that m is nowhere to be found in this final result, as it was long canceled out in the earlier steps right after applying Newton's second law for torque.

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Cheers!

- Andrew