Final answer:
To find the position of the pendulum at any time t, one must solve the differential equation representing the system's motion, involving a characteristic equation, complementary and particular solutions, and applying initial conditions.
Step-by-step explanation:
The student is tasked with solving the differential equation for a damped harmonic oscillator with a forcing function, which models the motion of a pendulum or a mass-spring system. The correct steps involve setting up the differential equation based on Newton's second law that represents the system's dynamics, finding the complementary solution by solving the associated homogeneous equation, and identifying a particular solution that corresponds to the forcing function. Once the complete solution is obtained, initial conditions can be applied to find the constants in the general solution, resulting in a description of the position of the pendulum at any given time t.
Steps to Solve the Differential Equation:
- Write down the differential equation that models the motion of the pendulum or mass-spring system.
- Solve the characteristic equation (r² + 0.04r + 16 = 0) associated with the complementary solution to the homogeneous differential equation.
- Determine the complementary solution using the roots of the characteristic equation.
- Find the particular solution to the differential equation with the given forcing function f(t) = 0.4 sin(5t).
- Add the complementary and particular solutions to obtain the general solution for the position of the pendulum.
- Utilize initial conditions to solve for any unknown constants in the general solution.
The equations of motion for a simple harmonic oscillator are derived from the restoring force being directly proportional to the displacement. The general form of the restoring force is F = -kx, where k is the force constant determined by k = mg/L for pendulum problems.