What is the Amplitude of a Pendulum?

In summary: However, by using a series approximation and making sure that the amplitude isnt too big, we can get a good approximation for the frequency of oscillations.
  • #1
potmobius
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0
How can one determine the amplitude, frequency and period of an amplitude? this is not homework, i was just curious, because i knew how to find the time using the 2 pi sqrt(l/g), but wanted to know about this, since i am learning about waves and harmonic motion! Help would be appreciated. Thanks!





P.S. this is my first time using physics forums, so tell me if i should change the way i ask a question or if i made any kind of mistake :P
 
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  • #2
Determine as in measure? Or as in calculate using Newton's laws?
 
  • #3
The [tex]2\pi \sqrt{\frac{l}{g}}[/tex] formula is only good for small amplitude displacements around 5 degrees, this is the harmonic approximation.

Lets look how we get this. Newton says, for the tangential component:

[tex]ml\ddot \varphi = - mg\sin\varphi[/tex]

Where [tex]\varphi[/tex] is the angle between the vertical and the string of the pendulum.

So the equation of motion for the pendulum:

[tex]\ddot\varphi = -\frac{g}{l}\sin\varphi[/tex]

Now as we see this is a non-linear differential equation. For small displacements (i.e. small angles) that is:
[tex]\sin\varphi \approx \varphi [/tex]

So:

[tex]\ddot\varphi = -\frac{g}{l}\varphi[/tex]

As we see this equation describes simple harmonic motion, and we can extract the frequency of oscillations:

[tex]\omega^2=\frac{g}{l} \Longrightarrow T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{l}{g}}[/tex]

So we obtained the formula for small displacements of the pendulum, and we can see that it (of course) doesn't depend on the amplitude...

Now if the displacements arent small we cannot approximate the sine like we did.

In this case after integrating the equation once and some manipulation, we obtain for the period:

[tex]T(\varphi_0)=4\sqrt{\frac{l}{g}}\int_0^{\frac{\pi}{2}} \frac{d\psi}{\sqrt{1-k^2\sin^2\psi}}[/tex]

Where [tex]k=\sin\left(\frac{\varphi_0}{2}\right)[/tex] Here \varphi_0 is the amplitude(maximum displacement) of the pendulum.

As we see this is an elliptic integral of the first kind. So the period of the pendulum at arbitrary amplitudes cannot be given using elementary functions.
We can however use a series approximation for the elliptic integral. Using this we get:

[tex]T(\varphi_0)=2\pi\sqrt{\frac{l}{g}}\sum_{n=0}^{\infty}\left[\frac{(2n-1)!}{(2n)!}\sin^n\frac{\varphi_0}{2}\right][/tex]

If the amplitude is still small but not that big, then we can further approximate the sines (now I do it upto fourth order of the amplitude):

[tex]T(\varphi_0)=2\pi\sqrt{\frac{l}{g}}\left[1+\frac{1}{16}\varphi_0^2+\frac{11}{3072}\varphi_0^4 +\dots\right][/tex]

So we can conclude that, we can't get an exact solution even for such trivial and simple configurations...
 

1. What is the definition of amplitude in a pendulum?

Amplitude in a pendulum refers to the maximum displacement of the pendulum bob from its equilibrium position. It is typically measured in degrees, radians, or meters.

2. How does the amplitude affect the period of a pendulum?

The amplitude of a pendulum does not affect its period. The period of a pendulum is solely determined by its length and the gravitational acceleration at its location.

3. What factors can affect the amplitude of a pendulum?

The amplitude of a pendulum can be affected by factors such as the length of the pendulum, the mass of the pendulum bob, the angle at which it is released, and the air resistance.

4. How can the amplitude of a pendulum be measured?

The amplitude of a pendulum can be measured by using a protractor to measure the angle of displacement from the equilibrium position. It can also be calculated using trigonometric functions if the length of the pendulum and the angle of displacement are known.

5. How does the amplitude of a pendulum change over time?

In a simple pendulum, the amplitude remains constant as long as there is no external force acting on it. However, in a damped or forced pendulum, the amplitude may decrease or vary over time due to external factors such as friction or periodic driving forces.

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