- Good morning.
The period of simple harmonic motion is the time it takes to complete one full cycle.
Flipping Physics
The symbol for period is capital 'T'
And the units for period are usually seconds or seconds per cycle; however, the units for period could also be
minutes, hours, days, fortnights, decades, millenniums....
Using our previously defined positions of 1 & 3,
where the object where the magnitude of the of the object's displacement
equals the amplitude which is the maximum displacement from equilibrium position,
and where position 2 is at the equilibrium position.
Recall that the simple harmonic motion pattern is
1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3,.. and so on.
Olivia, if the object starts at position 1, please describe one full cycle.
- If it starts at position 1, it goes through 2, then 3, then back to 2, then back to position 1 for one full cycle.
So, the time it takes to do that will be the period.
- Correct. Kevin, if the object starts at position 3, please describe one full cycle.
- So, that would be 3, 2, 1, 2, 3.
- Also correct. Anish, if the object starts at position 2, please describe one full cycle.
- It starts at 2, then … well, which way is it going?
- That's that's a good question. Let's have it moving toward position 3.
- Then it would go 2, 3, 2 … oh and then it needs to keep going to 1 and then back to 2 for one full cycle.
- Very nice! Notice if it were going the other way, the order would be 2,1,2,3,2
for one full cycle, and the period would be how long it takes for that to occur.
- Mr. P? - Yes, Olivia?
- What if the object starts somewhere between those three positions?
- Okay, sure. So, then one full cycle would be when the object starts and ends at the same location
and is moving in the same direction at the end as it was at the start.
You can see this is true for position 2. If it starts at position 2 and is moving to the right,
then the first time it gets back to position 2, it is moving to the left.
Which means it did not go through one full cycle;
it has to continue all the way back to position 2 and be moving to
the right before it has gone through one full cycle.
We can do the same thing with the object starting halfway between positions two and one
and moving to the right.
The first time it gets back to halfway between positions 1 & 2, it is moving to the left;
which means it has not gone through one full cycle,
it has to continue all the way back to halfway between positions 1 & 2
and be moving to the right before it has gone through one full cycle.
All right, let's move on to the equations.
The period of a mass spring system equals 2 pi times the square root of the mass attached to the spring
divided by the spring constant of the spring.
The period for a pendulum equals 2 pi times the square root of the pendulum length
divided by the acceleration due to gravity.
The pendulum length is defined as the distance from the center of suspension
to the center of mass of the pendulum.
The center of suspension is the location of the top fixed end of the pendulum.
Hi, I'm sorry. I need a come in for a minute.
Um. So, I need to mention that this period equation is for a simple pendulum specifically.
A simple pendulum is where there is a point mass attached to a string or rod of negligible mass.
In other words, all of the mass of the simple pendulum is concentrated
at the end of the pendulum at its center of mass.
Again, this is called a simple pendulum.
Okay, you can get back to that.
Those are the equations.
Now, let's identify what affects the periods of pendulums and mass spring systems.
Kevin, which of these two period equations has amplitude in it?
- Neither of the two period equations has amplitude in it.
- So, do you think amplitude effects period?
- Uh, no it looks like amplitude does not affect period.
- That is correct.
And you can see in these demonstrations that amplitude does not affect the time it takes for one full cycle.
Adjusting just the amplitude of the pendulum or the amplitude of the mass spring system
does not cause a change in the period of simple harmonic motion.
Anish, is acceleration due to gravity in the period equation for a mass spring system?
- No. So a change in the acceleration due to gravity does not affect the period of a mass-spring system.
- Whoa. So, a mass-spring system will have the same period on the Moon as it does on the Earth.
- Whoa!
- Olivia, is the mass of the pendulum bob in the equation for the period of a pendulum?
- Uh, who is Bob?
- Oh, I'm sorry. The the mass at the end of the pendulum is typically called a bob.
Have I not explained that before?
- No. I would remember Bob.
- Again, I am sorry. But, so, the question is does the mass of the pendulum bob affect the period?
- The mass of Bob is not in the equation, so no, his mass does not affect the period.
- Correct. As you can see in this demonstration,
changing just the mass of the pendulum Bob by having the pendulum Bob be wood aluminum or brass
does not have any effect on the period of the pendulum.
That is why all of these pendulums have the same period.
Anish, what if we increase the mass in a mass spring system?
What happens to the periods then?
- If we increase the mass in a mass-spring system …
well, the mass is in the numerator. So, an increase in mass will increase the period.
- Yes, as you can see an increase in mass increases the time it takes for one full cycle.
Olivia, what happens to the period of a mass spring system if we increase the spring constant?
- We are increasing the spring constant.
Okay, because the spring constant is in the denominator, increasing the spring constant will decrease the period.
- That's right. As you can see, an increase in spring constant decreases the time it takes for one full cycle.
Kevin, what happens to the period of a pendulum
if we increase the distance between the center of suspension and the center of mass of the pendulum?
- L, the pendulum length is on the top of the fraction,
so an increase in the pendulum length will increase the period of the pendulum.
- Again correct.
The longer the distance between the center of suspension and the center of mass of the pendulum,
the longer it takes for the pendulum to go through one full cycle.
Anish, if we increase the acceleration due to gravity, what happens to the period of a pendulum?
- The acceleration due to gravity is in the denominator;
so increasing the acceleration due to gravity will decrease the period of the pendulum.
- Yes. Now, I do apologize, however, the Flipping Physics budget does not allow for me to leave the surface
of the planet to demonstrate how changes
in the acceleration due to gravity affect the period of a pendulum.
I'm sorry!
- Maybe if you donated more!
- Kevin. I like where your head is at. Thanks!
Okay, I would like to take a moment to go back to the
amplitude of a pendulum not having an effect on the period of a pendulum.
Realize all of the amplitudes for these pendulums are less than 15 degrees.
- That’s right. A pendulum is only in simple harmonic motion if its amplitude is less than 15 degrees.
- Correct. I have now added a pendulum with an amplitude of roughly 45 degrees.
Notice how the period of this pendulum is not the same as the others.
That is because of this pendulum,
which has an amplitude greater than 15 degrees,
is no longer in simple harmonic motion, and therefore, the period
equation for simple harmonic motion no longer applies to this pendulum.
The motion of this pendulum is still periodic;
however, it is no longer simple harmonic motion.
Thank you very much for learning with me today.
I enjoyed learning with you.