Final Answer:
A) The equation for angular displacement (Θ) is given by Θ(t) = 0.2 cos(√(g/l) * t) + (30/√(3gl)) * sin(√(g/l) * t), where g is the acceleration due to gravity (approximately 9.8 m/s²), l is the length of the pendulum (0.6 m), and t is the time.
B) The first time (t₁) and angular position (Θ₁) where the potential energy equals 1/3 of the kinetic energy are t₁ ≈ 0.3 s and Θ₁ ≈ 0.161 rad.
Step-by-step explanation:
In the simple harmonic motion of a pendulum, the angular displacement (Θ) follows a standard equation. To derive the specific equation for this pendulum, we use the given initial conditions at t=0, where Θ=0.2 rad and v=30 cm/s. By solving for the constants in the general equation, we obtain the final expression for angular displacement.
For part B, the equality of potential energy (PE) and kinetic energy (KE) expressions is considered, incorporating the provided conditions. Solving this equation yields the time (t₁) when the energies are equal. Substituting t₁ back into the displacement equation provides the corresponding angular position (Θ₁).
These results offer a detailed insight into the pendulum's motion, providing both the equation for angular displacement and the specific time and position where potential and kinetic energies are in the specified ratio.