Final answer:
The equations of motion for a motor-driven pendulum involve harmonic motion formulas, including displacement as a function of time and the transformation of physical constants (velocity, spring constant) to terms applicable to pendulums. The system is analyzed through Newton's second law, small angle approximations, and the addition of a motor-driven force.
Step-by-step explanation:
Equations of Motion for a Motor-Driven Pendulum
The motion of a pendulum that is driven by a motor can be represented using the equations of harmonic motion. Utilizing Newton's second law, the force equation for a damped harmonic oscillator with the addition of a driving force F is used to analyze the system. For a simple pendulum, variables such as velocity (v) are expressed as v = Lω, where L is the length of the pendulum and ω is the angular velocity, the spring constant (k) corresponds to k = mg/L with m being the mass and g the acceleration due to gravity, and the displacement (x) is given by x = Lθ.
For small angles, where sin θ ≈ θ, the torque τ acting on the pendulum can be described by τ = -mglθ, considering the restoring force. This relationship results in a differential equation that governs the angular displacement θ over time, typically characterized by a second derivative of θ with respect to time and proportional to -θ. We can solve this to find the detailed behavior of the pendulum's motion, including its periodicity and amplitude, especially when it is in a state of forced, damped harmonic motion.
When considering a mass attached to a vertical spring and a rotating disk that applies a driving force, the displacement equation becomes more complex, involving the driving angular frequency ω and the driving force amplitude Fo. The equation of motion for such a system can include terms like Fo·sin(ωt) to account for the driving force provided by the motor.