Pendulum & SHM: Exploring the Relationship and Limits of Simple Harmonic Motion

In summary, a simple pendulum only exhibits simple harmonic motion (SHM) at small angles close to the equilibrium position because the restoring force is only approximately linear at those angles. As the amplitude increases, the period is no longer constant and the formula for period (i.e. constant independent of amplitude) no longer applies. This is due to the fact that for larger angles, the parallel acceleration is proportional to the sine of the angle, making the problem more difficult to solve. Additionally, at larger angles, the pendulum may start to vibrate slightly, indicating that it is no longer undergoing SHM alone.
  • #1
sexysam_short
Why is it that a simple pendulum only shows SHM at small angles close to the equilibrium position?
 
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  • #2
Because the restoring force is not linear and is only approximately linear at small amplitudes.
 
  • #3
Consider that if the launch angle of the bob is 180 degrees (the "string" would have to rigid to maintain the length), the bob would balance there. Effectively this means the period is infinite for amplitude 180 degrees. It is easy to see then that as the amplitude becomes larger, the period is no longer given by the simple formula (i.e. constant independent of amplitude), but will increase.
 
  • #4
If you resolve the gravitational force on a pendulum bob at angle θ to the vertical into a component perpendicular to the circular path (which just puts tension on the pendulum arm) and the component parallel to the path (which accelerates and decelerates the pendulum) you find that, from the right triangle set up, the parallel acceleration is proportional to sin(θ). For small values of θ that is very close to θ itself. It is that simplicity: acceleration= θ that gives "simple harmonic motion". For larger angles, we would have to use sin(&theta); instead of θ and that gives a much harder problem.
 
  • #5
Also, something I've noticed is when the angle is fairly large, the pendulum begins to slightly vibrate as it swings. This indicates it's no longer undergoing SHM alone.
 
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FAQ: Pendulum & SHM: Exploring the Relationship and Limits of Simple Harmonic Motion

What is a pendulum?

A pendulum is a weight suspended from a pivot point that can swing back and forth due to the force of gravity.

How does a pendulum demonstrate simple harmonic motion?

A pendulum demonstrates simple harmonic motion because its motion follows a sinusoidal pattern, with the pendulum swinging back and forth at a constant frequency and amplitude. This motion is governed by the restoring force of gravity.

What factors affect the period of a pendulum?

The period of a pendulum is affected by the length of the pendulum, the mass of the weight, and the acceleration due to gravity. Other factors such as air resistance and friction can also affect the period.

What is the relationship between a pendulum and simple harmonic motion?

A pendulum is a real-life example of simple harmonic motion. It follows the same mathematical equations and principles as other simple harmonic oscillators, such as a mass-spring system.

What are the limits of simple harmonic motion for a pendulum?

The limits of simple harmonic motion for a pendulum occur when the amplitude is too large or the motion is affected by external forces, such as air resistance or friction. In these cases, the pendulum's motion will deviate from a perfect sinusoidal pattern and may not follow the expected equations for simple harmonic motion.

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