Show that the potential energy of a simple pendulum is proportional

In summary, the potential energy of a simple pendulum can be expressed as U=mgl(1-cos(theta)), where theta is the angular displacement. In the small amplitude limit, where theta is relatively small, cos(theta) approaches 1, making the potential energy proportional to the square of the angular displacement.
  • #1
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Homework Statement



Show that the potential energy of a simple pendulum is proportional to the square of the angular displacement in the small amplitude limit.


Homework Equations



U=mgh
sin^2wt + cos^2wt=1


The Attempt at a Solution



I can't figure out where to start.
 
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  • #2
So nobody wants to help a man out? ic...thanks guys!
 
  • #3
Well, you haven't really given anybody anything that lends a hand towards helping you. What are your thoughts so far?
 
  • #4
Mindscrape said:
Well, you haven't really given anybody anything that lends a hand towards helping you. What are your thoughts so far?

Again, i don't know where to start on this one. I know that mgh=mgl(1-cos(theta)) but i don't know how to apply this in the problem. I'm not sure where to go.
 
  • #5
Where did the mgh=mgl(1-cos(theta)) come from?
 
  • #6
Even if you don't know where to start, you should at least state what you know. For example, you might know that a general oscillator will oscillate with a force [tex]F = -kx[/tex], and that potential energy is related to force by [tex] F = \frac{dU}{dt}[/tex]. Not to say that this helps you out, just the kind of thing that will help identify what level you are at, whether your class is calc based or not and such.

The potential you came up with is correct. What will happen to cos(theta) if theta is relatively small? What is the angular displacement for a simple pendulum?
 

Related to Show that the potential energy of a simple pendulum is proportional

1. How is potential energy related to a simple pendulum?

The potential energy of a simple pendulum is directly proportional to its height above the equilibrium point. This means that the higher the pendulum is lifted, the more potential energy it has.

2. What factors affect the potential energy of a simple pendulum?

The potential energy of a simple pendulum is affected by its mass, the gravitational acceleration, and the height of the pendulum above the equilibrium point. These factors determine the amount of potential energy the pendulum has at any given point in its swing.

3. How do you calculate the potential energy of a simple pendulum?

The potential energy of a simple pendulum can be calculated using the equation PE = mgh, where m is the mass of the pendulum, g is the gravitational acceleration, and h is the height of the pendulum above the equilibrium point.

4. Why is the potential energy of a simple pendulum considered a conservative force?

The potential energy of a simple pendulum is considered a conservative force because it is dependent only on the position of the pendulum and not on the path it takes to get there. This means that the amount of potential energy gained or lost by the pendulum will be the same regardless of the path it takes.

5. How does the potential energy of a simple pendulum change as it swings?

As a simple pendulum swings, its potential energy changes from maximum at the highest point of its swing (when it has the greatest height and potential energy) to minimum at the lowest point of its swing (when it has the lowest height and potential energy). It continuously exchanges potential energy for kinetic energy as it swings back and forth.

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