Angle/distance (Theta) moved by a pendulum bob in t seconds.

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Discussion Overview

The discussion revolves around modeling the motion of a pendulum in a programming project, focusing on the physics equations that govern its movement. Participants explore various equations and methods to calculate the angular displacement of the pendulum bob over time, addressing both theoretical and practical aspects of pendulum motion.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents initial equations for angular frequency and angular displacement, expressing uncertainty about the phase shift and the applicability of the equations for their programming model.
  • Another participant suggests starting from Newton's laws and provides a differential equation for the pendulum's motion, recommending a numerical integration method (Runge-Kutta) to solve for angular displacement over time.
  • A different participant discusses the small angle approximation, stating that for small angles, the sine function can be approximated by the angle itself, leading to a simpler differential equation.
  • One participant expresses gratitude for the guidance received and mentions successfully implementing the suggested equations, while also admitting difficulty in understanding the Runge-Kutta method due to a lack of background in differential equations.
  • Another participant warns that the small angle approximation may not yield accurate results for larger initial angles, suggesting that a non-linear approach may be necessary for more precise modeling.

Areas of Agreement / Disagreement

Participants generally agree on the importance of using differential equations to model pendulum motion, but there are competing views on the applicability of the small angle approximation and the methods to be used for larger angles. The discussion remains unresolved regarding the best approach for modeling pendulum motion at various angles.

Contextual Notes

Limitations include the dependence on the small angle approximation for certain equations, which may not be valid for larger initial angles. The discussion also highlights the need for a solid understanding of differential equations for accurate modeling.

PieKook
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Hello.

This is not a Physics homework problem, but rather a Programming project. I am asked to model a pendulum using the Java Swing class for graphics, but I'm having a few problems understanding the Physics behind it, and would appreciate a few answers and some guidance.


Doing the research, I found some energy equations (K = 1/2*m*v^2; U = mgh), but I don't think they'll be too useful in my calculations.

I then found the following equations:

Omega (W) = SQRT(g/L), where
Omega is the angular frequency,
g is the acceleration due to gravity, and
L is the length of pendulum the string.

Theta (Angular Frequency) = Theta_MAX * cos(Omega * dt + Phi), where
Theta is the angular distance moved in the given time period dt,
Theta_MAX is the maximum angle for the pendulum, and
Phi is the phase shift.

I think I want to use the Theta equation for program, but I'm not sure how.



Here are my initial assumptions for my model:

* Bob starts from rest at an angle of 20 degrees (0.349 radians)
* g = 10 m/s^2
* L = 100 m
* dt = 0.1 sec (time increment for displaying the location of the bob at different intervals)




Here are my calculations:

Omega = SQRT(g/L) = SQRT(10/100) = 0.32 rad/s (approximately 18.33 degrees per second)


Now here's where I think my calculations are incorrect when I try to find Theta, the angular displacement in each 0.1 second time increment. This angle will be used to draw the pendulum string (using the Java drawLine method).

Theta = Theta_Max * cos(Omega * dt + Phi)

Theta = 0.349 * cos(0.32 * 0.1 + 0.349) = 0.349 (?)

It seems that in this equation, as the time increment dt approaches to 0, the angular displacement Theta moved in these 0 seconds approaches to 0.349 * cos(Omega * 0 + Phi) = 0.349 * cos(Phi) = 0.349 * cos(0 + 0.349) = 0.328 rad, which does not make sense.


Am I mistaken with the phase shift? Or is the whole equation misplaced or something? I'm fairly sure this is the equation I need to figure out how far the bob has moved in each 0.1 second time increment.


Any information on the matter will be greatly appreciated.

Thank you.
 
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Did you read the wiki artical on pendulums? That is an improvement on what you currently have.

The best place to start a pendulum model is from Newtons laws.

http://Newton.ex.ac.uk/research/qsystems/people/sque/physics/simple-pendulum/" is a link to a site the gives a pretty good derviation. For your model you want to start from:

\ddot {\theta } = - \frac g l sin(\theta)

Use a 4th order Runga kutta integration scheme to get \theta as a function of time.
 
Last edited by a moderator:
As long as you are working with a pendulum whose length of swing is very short compared to the length of the pendulum itself, \theta will be small. For small angles sin(\theta) is approximately equal to \theta so the differential equation
\ddot {\theta } = - \frac g l sin(\theta)
can be approximated by
\ddot{\theta}= -\frac{g}{l} \theta
and the general solution to that is
\theta(t)= C cos(\sqrt{\frac{g}{l}}t)+ D sin(\sqrt{\frac{g}{l}}t)[/itex]<br /> In particular, if the pendulum is held up to angle \Theta_0 and released (initial speed 0) at t= 0, <br /> \theta(t)= \Theta_0 cos(\sqrt{\frac{g}{l}}t)[/itex]
 
Thank you very much for your time and help. I implemented it using this equation, and the pendulum works perfectly. :smile:



Integral said:
Did you read the wiki artical on pendulums? That is an improvement on what you currently have.

I did, as well as my old Physics book and a few articles Google retrieved for me, but none of them gave me the equation you guys had in mind. My search query was "Pendulum Motion Equations."



Integral said:
The best place to start a pendulum model is from Newtons laws.

http://Newton.ex.ac.uk/research/qsystems/people/sque/physics/simple-pendulum/" is a link to a site the gives a pretty good derviation. For your model you want to start from:

\ddot {\theta } = - \frac g l sin(\theta)

Use a 4th order Runga kutta integration scheme to get \theta as a function of time.


I kind of get the derivation by following the steps, but I'm afraid I've never heard of the Runge–Kutta method. I tried to research it a bit, but I've never had a Differential Equations class, so it's all going over my head.






HallsofIvy said:
As long as you are working with a pendulum whose length of swing is very short compared to the length of the pendulum itself, \theta will be small. For small angles sin(\theta) is approximately equal to \theta so the differential equation
\ddot {\theta } = - \frac g l sin(\theta)
can be approximated by
\ddot{\theta}= -\frac{g}{l} \theta
and the general solution to that is
\theta(t)= C cos(\sqrt{\frac{g}{l}}t)+ D sin(\sqrt{\frac{g}{l}}t)[/itex]<br /> In particular, if the pendulum is held up to angle \Theta_0 and released (initial speed 0) at t= 0, <br /> \theta(t)= \Theta_0 cos(\sqrt{\frac{g}{l}}t)[/itex]
&lt;br /&gt; &lt;br /&gt; &lt;br /&gt; Thank you for the step-by-step explanation. I wouldn&amp;#039;t have thought of that. This, in addition to Integral&amp;#039;s link, have been very helpful.&lt;br /&gt; &lt;br /&gt; &lt;br /&gt; Thanks again!
 
Last edited by a moderator:
Note that Halls solution involves a small angle approximation, this means that it will not give accurate positions if the initial angle is much greater then ~.1 radians. This is the usual approximation that is made. If you want meaningful results for large angles you will need to find a way do solve the non linear problem ( where I stopped).

To do any decent work in mathematical models Differential equations are essential. (just a warning! :smile: )
 

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