Questions: The Simple Pendulum

Question 1

Pendulum A has a 100g bob; pendulum B has an identical string length but a 400g bob. How do their periods compare under small-angle oscillations?

• A. Pendulum B has half the period of A, because a heavier bob oscillates faster
• B. Pendulum B has twice the period of A, because more mass means more inertia
• C. They have identical periods, because period is independent of mass
• D. Pendulum B has a slightly longer period due to increased air resistance from the larger bob

Answer: (C)

Period is mass-independent: T = 2π√(L/g) contains no mass term. A heavier bob experiences a proportionally stronger gravitational restoring force, but it also has proportionally greater inertia — both increase by the same factor, leaving the period unchanged. This is the same reason all objects fall at the same rate in a gravitational field (Galileo's insight): mass cancels from the equation of motion. The period depends only on string length L and local gravitational acceleration g.

Question 2

A pendulum is released from 45° amplitude. Which statement correctly describes how T = 2π√(L/g) applies to this situation?

• A. The formula gives the exact period, because it was derived from Newton's second law without approximation
• B. The formula underestimates the period, because the true restoring force sin θ > θ at large angles
• C. The formula overestimates the period, because sin θ < θ for non-small angles, making the true restoring force weaker than the approximation assumes
• D. The formula works for any amplitude up to 90°; it only fails beyond 90°

Answer: (C)

For θ > 0, sin θ < θ (in radians). The true restoring force −mg sin θ is therefore weaker than the approximated force −mgθ used to derive T = 2π√(L/g). A weaker restoring force means slower oscillation — longer period. So the small-angle formula underestimates the true period at large amplitudes. At 45°, the true period exceeds the formula's prediction by about 4.5%. This is why precision pendulum clocks keep amplitudes small — even a slight increase in swing amplitude causes the clock to run slow.

Question 3

Doubling the length of a simple pendulum increases its period by a factor of √2, not by a factor of 2.

• T. True
• F. False

Answer: True

T = 2π√(L/g), so T ∝ √L. If L doubles, the new period is T' = 2π√(2L/g) = √2 · 2π√(L/g) = √2 · T ≈ 1.414T. To double the period you would need to quadruple the length (since √4 = 2). This non-linear dependence — period scales as the square root of length — is a characteristic feature of the pendulum and an important counterexample to the naive expectation that 'doubling the length doubles the period.'

Question 4

The period formula T = 2π√(L/g) is exact for most oscillation amplitudes, as long as the pendulum is a true 'simple pendulum' (point mass on a massless string).

• T. True
• F. False

Answer: False

The formula is approximate — valid only for small angles (typically θ < 15° for better than ~1% accuracy). It follows from the small-angle approximation sin θ ≈ θ, which linearizes the equation of motion. The true equation of motion is α = −(g/L) sin θ, which is nonlinear. The true period for amplitude θ₀ involves an elliptic integral and depends on θ₀ — it grows with amplitude. At 30° the error is ~1.7%; at 45° ~4.5%; at 90° ~18%. A real pendulum clock run at large amplitude would lose time.

Question 5

Why does the period of a simple pendulum become amplitude-dependent for large swings, even though it is amplitude-independent for small ones?

Model answer: For small angles, sin θ ≈ θ, making the restoring force linear in displacement and the equation of motion identical to that of a harmonic oscillator. A linear restoring force produces amplitude-independent oscillation — this is the hallmark of simple harmonic motion. For large angles, sin θ < θ, so the restoring force is weaker than the linear approximation predicts. This nonlinearity means the effective 'spring constant' depends on amplitude: at larger swings, the restoring force is proportionally weaker, and the pendulum oscillates more slowly. The period grows with amplitude because the nonlinearity breaks the SHM approximation.

The key connection is to SHM: amplitude independence is a consequence of linearity. Any system with a strictly linear restoring force (F = −kx) oscillates at a fixed frequency regardless of amplitude. The pendulum achieves this only approximately, for small θ. Once the sinusoidal nonlinearity becomes significant, the frequency (and period) depends on how far the pendulum swings — larger swing, weaker effective restoring force per unit displacement, slower oscillation.