2. Derive the equation for the period T and frequency f for a simple pendulum ( 6 marks)

We start by considering a simple pendulum: a mass m attached to a light, inextensible string of length l, swinging under the influence of gravity. 1. Force Analysis and Equation of Motion When the pendulum is displaced by an angle θ from the vertical, the restoring force along the arc is given by   F = –m·g·sinθ where g is the acceleration due to gravity. Since the motion is along a circular arc, we analyze the angular motion. The tangential acceleration aₜ relates to the angular acceleration θ¨ by:   aₜ = l·θ¨ Using Newton’s second law (F = m·a):   m·l·θ¨ = –m·g·sinθ We can simplify by canceling m from both sides:   l·θ¨ = –g·sinθ        (1) 2. Small-Angle Approximation and Simple Harmonic Motion For small angles (θ measured in radians), the approximation sinθ ≈ θ is valid. Substituting into (1):   l·θ¨ ≈ –g·θ Rearrange to obtain:   θ¨ + (g/l)·θ = 0       (2) Equation (2) is the standard form of the simple harmonic oscillator, whose general solution is:   θ(t) = θ₀ cos(ωt + φ) with angular frequency ω given by:   ω² = g/l   ⇒  ω = √(g/l) 3. Derivation of the Period T The period T of a simple harmonic oscillator is related to ω by:   T = 2π/ω Substituting ω:   T = 2π/√(g/l) = 2π√(l/g) 4. Derivation of the Frequency f The frequency f is the reciprocal of the period:   f = 1/T = 1/(2π√(l/g)) = (1/(2π))√(g/l) Summary: • Period:  T = 2π√(l/g) • Frequency: f = (1/(2π))√(g/l) This derivation assumes that the angle of oscillation is small so that the approximation sinθ ≈ θ holds, which allows the pendulum's motion to be described as simple harmonic motion.