Equation of motion for a simple pendulum

[lina29]

1. Write the equation of motion for the pendulum for different lengths. Use the motion at its greatest extension when t=0 and find the phase shift.

I am supposed to find the equation of motion (x=xmax cos($$\omega$$t+$$\phi$$) where xmax is the amplitude, $$\omega$$ is the angular frequency, and $$\phi$$ is the phase shift of the motion.

For my experiment the length was 102 cm, the amplitude was 35, the time for 25 swings was 50.8, the period was 20.271, and the angular frequency was .309

Based upon my data I found the equation to be x=35cos(.309t+$$\phi$$)

I'm just confused on how I would get the equation. Would I leave t as a variable or input t as 50.8/25. Also how would I find the phase shift?

 

[chaoseverlasting]

If 25 swings take 50.8 seconds, then your time per swing is 50.8/25. Your time period comes out to be twice that because two swings is one complete oscillation.

You use the time period to get the angular frequency.

 

[lina29]

I already found out the angular frequency I was just wondering how to find the phase shift

 

[olivermsun]

Look at your pendulum equation carefully. The problem tells you to choose t=0 at a moment when x is its greatest value, that is, when x=xmax. If so, what does cos(...) have to be, and what has to be inside the ( ) to make it that value?

 

[rwisz]

The equation of motion for a simple pendulum is given by x = xmax cos(\omega t + \phi), where x is the displacement of the pendulum, xmax is the amplitude, \omega is the angular frequency, t is the time, and \phi is the phase shift.

To find the equation for different lengths, you would need to calculate the value of \omega for each length using the formula \omega = \sqrt{\frac{g}{l}}, where g is the acceleration due to gravity and l is the length of the pendulum.

In your experiment, the length of the pendulum is 102 cm, so you would calculate \omega as:

\omega = \sqrt{\frac{9.8 \text{ m/s}^2}{0.102 \text{ m}}} = 3.09 \text{ rad/s}

Now, to find the phase shift, you can use the data you have collected for the time and period. The period of a pendulum is given by T = 2\pi\sqrt{\frac{l}{g}}, so we can rearrange this equation to find the phase shift, \phi:

\phi = \arcsin\left(\frac{t}{T}\right) - \omega t

Substituting in the values from your experiment, we get:

\phi = \arcsin\left(\frac{50.8 \text{ s}}{20.271 \text{ s}}\right) - (3.09 \text{ rad/s})(50.8 \text{ s}) = 0.07 \text{ rad}

Therefore, the equation of motion for your pendulum with a length of 102 cm would be:

x = 35 \cos(3.09t + 0.07)

Note that the value of t in the equation is still a variable, as it represents the time at any given point during the pendulum's motion. The value of 50.8/25 that you mentioned in your data is the average time for 25 swings, but it does not represent a specific point in time during the motion of the pendulum.