The Relationship Between Amplitude and Period for a Spring | scatter chart made by Burgs | plotly
David explains what affects the period of a mass on a spring (i.e. mass and spring constant. Amplitude, Period, Frequency, and Wavelength of Periodic Waves. This video is a continuation of The Relationship between Wave Frequency, Period, Wavelength, and Velocity .. Calculate the Period and the Wave Velocity of a Toy Spring. We have already seen that the stretch of a spring is proportional to the force is a simple harmonic oscillator, we know the amplitude does not affect the period.
Maybe it depends on the amplitude, so let's just check. If I asked you, if I asked you, if I pulled this back farther, if I increase the amplitude farther, will that change the period of this motion?
So, let's think about it. Some of you might say, yes, it should increase the period because look, now it has farther to travel, right?
Instead of just traveling through this amount, whoa that looked horrible, instead of just traveling through this amount back and forth, it's gotta travel through this amount back and forth. Since it has farther to travel, the period should increase. But some of you might also say, wait a minute. If we pull this mass farther, we know Hooke's law says that the force is proportional, the force from the spring, proportional to the amount that the spring is stretched.
So, if I pulled this mass back farther, there's gonna be a larger force that's gonna cause this mass to have a larger velocity when it gets to you, a larger speed when it gets to the equilibrium position, so it's gonna be moving faster than it would have. So, since it moves faster, maybe it takes less time for this to go through a cycle. But it turns out those two effects offset exactly.
In other words, the fact that this mass has farther to travel and the fact that it will now be traveling faster offset perfectly and it doesn't affect the period at all.
This is kinda crazy but something you need to remember. The amplitude, changes in the amplitude do not affect the period at all.
So pull this mass back a little bit, just a little bit of an amplitude, it'll oscillate with a certain period, let's say, three seconds, just to make it not abstract. And let's say we pull it back much farther. It should oscillate still with three seconds. So it has farther to travel, but it's gonna be traveling faster and the amplitude does not affect the period for a mass oscillating on a spring.
This is kinda crazy, but it's true and it's important to remember. This amplitude does not affect the period. In other words, if you were to look at this on a graph, let's say you graphed this, put this thing on a graph, if we increase the amplitude, what would happen to this graph? Well, it would just stretch this way, right? We'd have a bigger amplitude, but you can do that and there would not necessarily be any stretch this way.
If you leave everything else the same and all you do is change the amplitude, the period would remain the same.
The period this way would not change. So, changes in amplitude do not affect the period. So, what does affect the period? I'd be like, alright, so the amplitude doesn't affect it, what does affect the period?
Well, let me just give you the formula for it. So the formula for the period of a mass on a spring is the period here is gonna be equal to, this is for the period of a mass on a spring, turns out it's equal to two pi times the square root of the mass that's connected to the spring divided by the spring constant. That is the same spring constant that you have in Hooke's law, so it's that spring constant there. It's also the one you see in the energy formula for a spring, same spring constant all the way.
This is the formula for the period of a mass on a spring. Now, I'm not gonna derive this because the derivations typically involve calculus. If you know some calculus and you want to see how this is derived, check out the videos we've got on simple harmonic motion with calculus, using calculus, and you can see how this equation comes about.
But for now, I'm just gonna quote it, and we're gonna sort of just take a tour of this equation. So, the two pi, that's just a constant out front, and then you've got mass here and that should make sense.
Why does increasing the mass increase the period? Look it, that's what this says. If we increase the mass, we would increase the period because we'd have a larger numerator over here.
How does mass affect simple harmonic motion? + Example
That makes sense 'cause a larger mass means that this thing has more inertia, right. Increase the mass, this mass is gonna be more sluggish to movement, more difficult to whip around.
If it's a small mass, you can whip it around really easily. If it's a large mass, very mass if it's gonna be difficult to change its direction over and over, so it's gonna be harder to move because of that and it's gonna take longer to go through an entire cycle. This spring is gonna find it more difficult to pull this mass and then slow it down and then speed it back up because it's more massive, it's got more inertia.
As discussed above, the spring force varies in magnitude and in direction. Its magnitude can be found using Hooke's law. Its direction is always opposite the direction of stretch and towards the equilibrium position.
As the air track glider does the back and forth, the spring force Fspring acts as the restoring force. It acts leftward on the glider when it is positioned to the right of the equilibrium position; and it acts rightward on the glider when it is positioned to the left of the equilibrium position.
Let's suppose that the glider is pulled to the right of the equilibrium position and released from rest. The diagram below shows the direction of the spring force at five different positions over the course of the glider's path. As the glider moves from position A the release point to position B and then to position C, the spring force acts leftward upon the leftward moving glider.
As the glider approaches position C, the amount of stretch of the spring decreases and the spring force decreases, consistent with Hooke's Law.
Despite this decrease in the spring force, there is still an acceleration caused by the restoring force for the entire span from position A to position C.
At position C, the glider has reached its maximum speed. Once the glider passes to the left of position C, the spring force acts rightward. During this phase of the glider's cycle, the spring is being compressed. The further past position C that the glider moves, the greater the amount of compression and the greater the spring force.
This spring force acts as a restoring force, slowing the glider down as it moves from position C to position D to position E. By the time the glider has reached position E, it has slowed down to a momentary rest position before changing its direction and heading back towards the equilibrium position.
During the glider's motion from position E to position C, the amount that the spring is compressed decreases and the spring force decreases.
There is still an acceleration for the entire distance from position E to position C. Now the glider begins to move to the right of point C. As it does, the spring force acts leftward upon the rightward moving glider.
This restoring force causes the glider to slow down during the entire path from position C to position D to position E. Sinusoidal Nature of the Motion of a Mass on a Spring Previously in this lessonthe variations in the position of a mass on a spring with respect to time were discussed.
At that time, it was shown that the position of a mass on a spring varies with the sine of the time. The discussion pertained to a mass that was vibrating up and down while suspended from the spring. The discussion would be just as applicable to our glider moving along the air track. If a motion detector were placed at the right end of the air track to collect data for a position vs. Position A is the right-most position on the air track when the glider is closest to the detector. The labeled positions in the diagram above are the same positions used in the discussion of restoring force above.
You might recall from that discussion that positions A and E were positions at which the mass had a zero velocity. Position C was the equilibrium position and was the position of maximum speed.
Amplitude and period relationship
If the same motion detector that collected position-time data were used to collect velocity-time data, then the plotted data would look like the graph below. Observe that the velocity-time plot for the mass on a spring is also a sinusoidal shaped plot. The only difference between the position-time and the velocity-time plots is that one is shifted one-fourth of a vibrational cycle away from the other.
Also observe in the plots that the absolute value of the velocity is greatest at position C corresponding to the equilibrium position. The velocity of any moving object, whether vibrating or not, is the speed with a direction. The magnitude of the velocity is the speed.
Oscillation amplitude and period
The direction is often expressed as a positive or a negative sign. In some instances, the velocity has a negative direction the glider is moving leftward and its velocity is plotted below the time axis. In other cases, the velocity has a positive direction the glider is moving rightward and its velocity is plotted above the time axis.4 Simple Harmonic Motion Derivation of the Time Period for a spring mass oscillator
You will also notice that the velocity is zero whenever the position is at an extreme. This occurs at positions A and E when the glider is beginning to change direction. So just as in the case of pendulum motionthe speed is greatest when the displacement of the mass relative to its equilibrium position is the least.
And the speed is least when the displacement of the mass relative to its equilibrium position is the greatest.