Length/Time Period squared Relationship of a Simple Pendulum

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The discussion centers on deriving the relationship between the length and time period of a simple pendulum, specifically the constant g/(2π)². A user seeks clarification on the derivation, particularly regarding the differential equation and its solution. The response explains that the angular acceleration is represented by the double dot notation and that the parameters A₀ and φ in the solution correspond to the maximum angle and phase shift, respectively. The explanation emphasizes that the second derivative of the sine function leads to a characteristic frequency, which is fundamental in understanding simple harmonic motion. The conversation concludes with a note on using LaTeX for proper symbol representation.
Thinker8921
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I know the L/T^2 relationship of a simple pendulum gives a constant- g/(2pi)^2.
Could anyone please show me how it is derived?
Thanks in advance.
 
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Thinker8921 said:
I know the L/T^2 relationship of a simple pendulum gives a constant- g/(2pi)^2.
Could anyone please show me how it is derived?
Thanks in advance.

The force on the pendulum bob is:

F = -mgsin\theta = ma = mL\ddot\theta

For small angles, sin\theta \approx \theta so you have the differential equation:

\ddot\theta = -\frac{g}{L}\theta the solution of which is:

\theta = A_0\sin(\omega t + \phi) where \omega^2 = g/L with \omega = 2\pi/T

AM
 
Thankyou for the reply. I think I should have been a little clearer. I am in high school and this depth is a little new so I don't understand the whole explanation.
-The theta sign with the 2 dots*, I am thinking it is angular acceleration? That way, mLa will be torque of the pendulum bob.
- I get the differential, however not the next step with the A zero sign and phi sign. What do they represent. Is it (Started calculus a week ago).
Please could you clear my doubts.
Thanks again.
*How do I insert the actual symbols in this?
 
Thinker8921 said:
Thankyou for the reply. I think I should have been a little clearer. I am in high school and this depth is a little new so I don't understand the whole explanation.
Welcome to PF, by the way.
-The theta sign with the 2 dots*, I am thinking it is angular acceleration? That way, mLa will be torque of the pendulum bob.
That's right.
- I get the differential, however not the next step with the A zero sign and phi sign. What do they represent. Is it (Started calculus a week ago).
These are parameters for the initial condition. The A0 (I could have said \theta_0) is the maximum value of \theta, which occurs when the sin term = 1. The \phi is simply to adjust for the phase - ie. when the amplitude maximum occurs in relation to t. For example if amplitude was maximum at t=0, you would set \phi = \pi/2 so that the sin term gave a value of 1 (which is the maximum value for sin).

The idea here is that if you take the second derivative of \theta_0\sin (\omega t + \phi) you get -\omega^2\theta_0\sin (\omega t + \phi), which is simply -\omega^2 x the original function. Generally if the second derivative is -\omega^2 multiplied by the original function, the function must be a some sort of sine wave with frequency \omega. You will learn how to solve this kind of equation when you study differential equations.

*How do I insert the actual symbols in this?
You have to use Latex. See https://www.physicsforums.com/showthread.php?t=8997" for help on Latex.

AM
 
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Thanks, this has made it clearer to me.
 
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