§ 3. Simple & Spring Pendulums (Grade 9–10)
Comparison of periods T = 2π√(L/g) and T = 2π√(m/k). Harmonic oscillations involve periodic conversion between potential energy of displacement and kinetic energy of motion.
Theory Summary & Scientific Mechanism
Simple harmonic motion arises whenever a restoring force is proportional to the displacement from equilibrium: Fₓ = -kx. In a simple gravity pendulum, the restoring force is the tangential component of weight F_τ = -mgsin θ ≈ -(mg)/(L)s (for small angles θ ≪ 1 rad), whereas in a spring-mass oscillator it is Hooke’s elastic force F = -ky.
The damped oscillator obeys x + 2βx + ω₀² x = 0. For the simple pendulum, the natural angular frequency ω₀ = √(g/L) and Huygens’ period T₀ = 2π√(L/g) depend solely on string length L and gravity g (isochronism). Conversely, for the spring pendulum, ω₀ = √(k/m) and T₀ = 2π√(m/k) depend on mass m and stiffness k, remaining independent of gravity.
Twice during every oscillation period, energy converts completely between potential energy Eₚ at the turning points and kinetic energy Eₖ at equilibrium. With damping (β > 0), the amplitude decays exponentially as A(t) = A₀ e⁻β t, transferring mechanical energy into thermal dissipation Q.
Everyday Analogy
Think of a playground swing or a grandfather clock! Whether a child or an adult sits on the swing, the time of one back-and-forth swing depends only on the rope length L, not on the person’s mass.
Interactive Question & Derivation
How do the small-oscillation periods of a simple pendulum Tₚend and a spring pendulum Tₛpr change if the bob mass m is quadrupled (4×)?
- A: The period of both pendulums increases by a factor of 2
- B: The simple pendulum period remains unchanged, while the spring pendulum period doubles (2×)
- C: The simple pendulum period doubles, while the spring pendulum period stays unchanged
Study Hint: Look at Huygens’ formula T = 2π√(L/g): does the bob mass m appear in it, and how does √(L) change when length L increases?
Proof & Explanation: For a simple pendulum, both the gravitational restoring force and inertia are proportional to mass m, so Tₚend = 2π√(L/g) is independent of mass! In contrast, Hooke's restoring force F = -kx in a spring does not depend on mass, so quadrupling m doubles the period Tₛpr = 2π√(m/k).
Key Equations & 3D Simulation Parameters
θ + 2γθ + (g)/(L)sin θ = 0, y + 2γy + (k)/(m)y = 0
- String length L: 1.8 … 4.8 m (default: 3.5 m)
- Initial amplitude θ₀: 10 … 75 ° (default: 35 °)
- Bob mass m: 0.5 … 5 kg (default: 2 kg)
- Spring stiffness k: 5 … 35 N/m (default: 14 N/m)
- Damping coeff. γ: 0 … 0.25 s⁻¹ (default: 0.02 s⁻¹)