Part of a simple harmonic motion playlist (pendulums and simple harmonic oscillator): • Simple harmonic oscill...
Uncoupled simple pendulum: period of one complete cycle derivation, the pendulum simple harmonic motion.
In this derivation of simple pendulum small oscillations period, we start by decomposing the force of gravity into components, then we compute the torque exerted by gravity and write down the rotational equivalent to Newton's Second Law: tau=I*alpha. Because alpha is the second time derivative of angle, this is actually a second order differential equation, but unfortunately it's a non-linear differential equation! We expand sin(theta) in a power series and we have to make a small angle approximation to justify truncating the series in order to linearize the differential equation. We compute the error committed in this approximation, and it's only about 1% when the angle is less than 15 degrees.
After linearizing the differential equation with the sine expansion for a pendulum, we can guess the solutions to the simple harmonic oscillations, and we verify the simple pendulum equation of motion by differentiating twice to show that it obeys the differential equation. Then we obtain the formula for the period of oscillation of a simple pendulum. After we derive simple pendulum period, we apply the pendulum period formula to a simple example of oscillation physics.
Негізгі бет Derivation of simple pendulum period, equation of motion and example. Sine expansion for a pendulum.
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