- Good morning.
Kevin, remind me, what is required for an object to be in simple harmonic motion?
Flipping Physics
- Well, there has to be a restoring force, right?
- Right, the force which is always directed toward rest position.
- And the restoring force has to be proportional to the displacement from rest position.
- Correct, we have shown that a mass-spring system is in simple harmonic motion.
Now, we are going to show when a pendulum is in simple harmonic motion.
Let's start by showing a pendulum and a mass spring system simultaneously.
You can see for both the mass spring system and the pendulum, we have the same positions we had before.
Position 1 is where the mass is displaced to the right of equilibrium position,
and the magnitude of that distance is the amplitude of the motion.
Position 2 is where the mass is located at the equilibrium position.
Position 3 is where the mass is displaced to the left of equilibrium position.
And the magnitude of that distance is, again, the amplitude of the motion.
It is important to realize that x,
the displacement from equilibrium position in a mass spring system, is a linear displacement.
However in a pendulum, it is an angular displacement .
- Does that mean the units for the angular displacement for a pendulum are in degrees?
- Or they could be in radians, right?
- We should probably use the symbol theta to indicate the angular displacement for a pendulum instead of x.
- Correct,
theta is the symbol for angular displacement of a pendulum in simple harmonic motion
and the units can be in degrees or radians.
However, I will point out that when we work with the trigonometric equations for simple harmonic motion,
theta does need to be in radians.
We're not there yet; however, we will be a bit later.
Olivia, please tell me what we currently know about the magnitudes of the velocities
of the objects in simple harmonic motion at positions 1, 2, and 3.
- Let’s see … at the two extreme positions, 1 and 3,
where the magnitude of the displacement from rest position equals the amplitude,
the object changes direction, so the velocity is zero.
At rest position, position 2, the magnitude of the velocity of the object is at a maximum value.
- Correct. Anish, what about the magnitudes of the accelerations at those positions?
- Sure. At rest position, or position 2, the displacement from equilibrium position is zero,
so the restoring force is zero, so the acceleration of the object is also zero.
At positions 1 and 3, the displacement from equilibrium position has its maximum magnitude,
which is the amplitude,
so the restoring force has its maximum magnitude,
so the acceleration of the object also has its maximum magnitude.
- That is all correct.
So, we've shown that a pendulum appears to have
the same characteristics of simple harmonic motion as a mass spring system;
however, we have yet to discuss when a pendulum is in simple harmonic motion.
In order to do so, we need to identify what the restoring force is for a pendulum in simple harmonic motion.
- Was there a question there? I feel like there was no question.
- Yeah, is there a question?
- Okay, fair enough. Anish what is the restoring force for a pendulum in simple harmonic motion?
- There are only two forces acting on the pendulum;
the force of gravity down and the force of tension in the string acting along the string. It has to be one of those.
Is the restoring force the force of gravity?
- No. The force of gravity is constant so it is not proportional to displacement from equilibrium position.
- Oh, it has to be the force of tension then.
- It cannot be the force of tension.
The force of tension has its greatest magnitude when the pendulum is at the equilibrium position.
- It does?
- Yeah, remember when Mr.p spun the bucket in a vertical circle at a constant angular velocity?
The maximum force of tension in the rope was when the bucket was at the bottom of its swing
and the same would be true here.
- If it is not the force of gravity or the force of tension,
then I have no idea what the restoring force for a pendulum is.
- Was there a question there? I feel like there was no question.
- Okay. Uh. Mr.p, what is the restoring force for a pendulum in simple harmonic motion?
- That’s the question I asked you.
- Right. But I have no idea what the answer is.
- Okay. That’s fine. I can do it. So,
The restoring force for a pendulum in simple harmonic motion is the component
of the force of gravity which is in the tangential direction.
This fits the criteria for a restoring force for simple harmonic motion
because it is directed toward the equilibrium position
and is proportional to the displacement from equilibrium position.
Force of gravity tangential to the motion equals mass times acceleration due to gravity
times the sine of the angle the pendulum makes with the vertical.
When that angle is zero degrees,
that makes the sine of the angle equal to zero
and therefore the force of gravity tangential is also equal to zero.
As the angle increases, the force of gravity tangential increases.
In other words, as the displacement from equilibrium position increases,
the force of gravity tangential increases.
Okay,
notice how the force of gravity tangential starts out pointed toward rest position,
and, as the angular displacement from rest position increases,
the force of gravity tangential points increasingly to a location which is below the equilibrium position.
This means the force of gravity tangential is only a restoring force for small angles.
Typically a pendulum is considered to be in simple harmonic motion
as long as the amplitude of the motion is less than 15 degrees.
- Wait, where did 15 degrees come from?
- Yeah, uh, 15 degrees?
- We won't get into the calculus behind
why the maximum amplitude for simple harmonic motion of a pendulum is 15 degrees.
However, know that it is because of what is called the small angle approximation.
The small angle approximation states that using radians as the dimensions for theta, the sine of theta is
approximately equal to theta for small angles.
Again, that would be for angles less than roughly 15 degrees.
I will point out that some sources say that it has to be less than 10 degrees; however,
I think that a 15 degree maximum amplitude provides a reasonable approximation of simple harmonic motion.
Thank you very much for learning with me today.
I enjoyed learning with you.