(II) A simple pendulum is 0.30 m long. At t=0 it is released from rest starting at an angle of 1

(II) A simple pendulum is 0.30 m long. At t=0 it is released...

(II) A simple pendulum is 0.30 $\mathrm{m}$ long. At $t=0$ it is released from rest starting at an angle of $13^{\circ} .$ Ignoring friction, what will be angular position of the pendulum at (a) $t=0.35 \mathrm{s},$ (b) $t=3.45 \mathrm{s},$ and $(c) t=6.00 \mathrm{s?}$

Key Concepts

Simple Pendulum

A simple pendulum consists of a mass suspended from a fixed point by a lightweight, inextensible string or rod, free to swing under the influence of gravity. Its motion, when displaced from the equilibrium, demonstrates periodic, or oscillatory, behavior and is a classical example used to study dynamics in a gravitational field.

Harmonic Oscillator

The motion of a simple pendulum, especially under small displacement conditions, can be modeled as a simple harmonic oscillator. This means that its restoring force is directly proportional to the displacement, leading to sinusoidal oscillations that are characterized by predictable periodic motions.

Angular Frequency

Angular frequency is a measure of how quickly an oscillator completes its cycles and is a fundamental parameter in describing harmonic motion. In the context of a pendulum, it is determined by the gravitational acceleration and the length of the pendulum, influencing the period and phase of the oscillations.

Initial Conditions and Phase Angle

The analysis of pendulum motion often requires incorporating initial conditions, such as the initial displacement and velocity, to accurately determine the phase of the oscillation. This phase angle indicates the state of the motion at t=0 and is essential for predicting the pendulum's angular position at any later time.

Small-Angle Approximation

For angular displacements that are relatively small (typically less than about 15 degrees), the sine of the angle is approximately equal to the angle in radians. This approximation simplifies the equations of motion for a pendulum, allowing it to be analyzed using the framework of simple harmonic motion.