Equation for simple harmonic oscillators | Physics | Khan Academy

Summary

The video explains how to represent the motion of a simple harmonic oscillator using a sine or cosine function. It demonstrates how to determine the amplitude, period, and phase shift based on the oscillator's graph. The instructor details how to write the equation for position as a function of time, taking into account amplitude (maximum displacement) and period (time for one full cycle). The video covers when to use sine, cosine, or their negative counterparts depending on the starting position of the graph, helping users understand how to model harmonic motion mathematically.

Takeaways

• 📊 The motion of a simple harmonic oscillator can be represented on a graph, with the amplitude showing the maximum displacement from equilibrium and the period indicating the time it takes to complete a cycle.
• 🌀 The equation representing a simple harmonic oscillator's motion needs to reflect the position of the mass as a function of time, either using sine or cosine functions.
• 📈 The choice between sine or cosine depends on where the graph starts. If the motion starts at a maximum, cosine is used; if it starts at zero or equilibrium, sine is used.
• ⚙️ The amplitude (A) represents the maximum displacement and scales the cosine or sine function so the oscillator reaches its actual maximum displacement.
• ⏳ The period (T) represents how long it takes for the oscillator to complete one full cycle. It is essential to adjust the equation to reflect the actual period.
• 🧮 The angular frequency (ω), which equals 2π divided by the period, ensures the function repeats correctly for any given period of the oscillator.
• 🔄 Simple harmonic motion can be visualized as circular motion on a unit circle, which helps in understanding cyclic processes in a broader context.
• 🧑‍🏫 To calculate the position at any given time, plug in the desired time (t) into the equation, which then returns the corresponding position value.
• 🔀 If the oscillator starts at a minimum, a negative cosine function is used; if it starts at equilibrium and goes down, a negative sine function is used.
• 📝 The equation for simple harmonic motion always follows the form: X(t) = ±Amplitude × (sine or cosine) × (2π/Period) × Time.

Q & A

What is the amplitude in the context of a simple harmonic oscillator?

The amplitude is the maximum displacement from equilibrium in the motion of the simple harmonic oscillator. It is represented as the highest value the position graph reaches.

What is the period of a simple harmonic oscillator?

The period (T) is the time it takes for the oscillator to complete one full cycle and return to its starting point. This can be measured from peak to peak, trough to trough, or any analogous point in the cycle.

Why is cosine used in the equation for a simple harmonic oscillator when the motion starts at a maximum?

Cosine is used because at time t = 0, the value of cosine is 1, representing the maximum displacement of the oscillator. If the motion starts at a maximum, cosine provides the appropriate mathematical representation.

How do you modify the equation if the amplitude is not 1?

If the amplitude is not 1, you multiply the cosine function by the amplitude (A). This ensures that the maximum displacement is correctly represented, as cosine alone only reaches a maximum of 1.

What role does the angular frequency (omega, ω) play in the equation?

Angular frequency (ω) is used to control the period of the oscillation. It adjusts the frequency at which the motion resets, allowing the equation to reflect any period, not just the default period of 2π.

How do you adjust the equation to reflect a specific period?

To adjust the equation for a specific period, you multiply time (t) by ω, where ω = 2π/T (T being the period). This ensures that the function resets after each period T, as desired.

Why do we use radians instead of degrees in the equation?

Radians are used because they simplify the relationship between angular frequency and the period of the oscillator. In physics, radians are the standard for measuring angles in trigonometric functions like sine and cosine.

How does the graph of the motion change if the period increases?

If the period increases, the graph stretches horizontally, meaning it takes more time to complete one cycle. The amplitude remains the same unless specified otherwise.

When would you use sine instead of cosine in the equation?

Sine is used if the motion starts at zero (equilibrium) and moves up. If the motion starts at equilibrium but moves down, you would use negative sine.

How do you handle an oscillator that starts at a minimum value?

If the motion starts at a minimum value, you use negative cosine in the equation. This reflects that the motion begins at the lowest displacement instead of the highest.

Outlines

00:00

📊 Understanding Simple Harmonic Motion Graph

The instructor explains how to represent the motion of a simple harmonic oscillator using a horizontal position graph, which shows the amplitude as the maximum displacement from equilibrium. The period (T) represents the time taken for one complete cycle, which can be measured from peak to peak or trough to trough. The graph can represent different oscillators by stretching vertically or horizontally to adjust amplitude or period. The need for an equation to describe this motion is also introduced, focusing on how time relates to position in this context.

05:01

🧠 Choosing the Right Trigonometric Function

The paragraph focuses on selecting the appropriate trigonometric function (sine or cosine) to model the motion. Since the motion starts at a maximum, cosine is chosen because cosine at T = 0 is 1, representing the maximum displacement. However, just using cosine is insufficient because the amplitude might not be 1, so multiplying by the amplitude ensures the equation accurately reflects the motion. A real-world example is introduced, where the amplitude is 0.2 meters, showing how this scales the graph appropriately.

10:01

⏱ Adjusting for Time and Period

The explanation shifts to why time (T) alone isn't sufficient to represent the motion correctly. The need to incorporate angular frequency (omega) is introduced to adjust the period, ensuring the function resets after the correct time (not always two pi seconds). Omega helps tune the period to match real-world conditions, like six seconds in the example. Angular velocity, though related to circular motion, applies here because cyclic processes (like oscillators) can be represented on a unit circle, linking angular frequency to the period.

🔄 Deriving the Equation for Harmonic Motion

The paragraph demonstrates how to derive a complete equation for simple harmonic motion by introducing the angular frequency as two pi over the period (T). The resulting function will reset at the appropriate times, ensuring the graph accurately models the oscillator’s motion. A breakdown of the equation shows how at T = 0, the cosine gives the maximum value, and at T = the period, the function resets. The example uses a period of six seconds and amplitude of 0.2 meters, resulting in an accurate representation of the oscillator's movement.

🌀 Simplifying and Solving for Time-Dependent Position

In this section, the instructor emphasizes how to apply the formula to solve real problems. For instance, to find the position at a specific time (e.g., 9 seconds), one would plug the value into the function. The formula then calculates the position at that moment. The instructor assures that this process becomes quicker with practice, with the key being the use of sine or cosine based on whether the motion starts at a maximum, minimum, or equilibrium.

📈 Different Scenarios for Using Cosine or Sine

The paragraph offers practical advice on choosing between cosine and sine for different starting points in oscillation. Cosine is used when the graph starts at a maximum or minimum, while sine is used when the motion starts at zero (equilibrium). The instructor provides a detailed example with a period of four seconds and amplitude of three meters, and shows how to adjust for cases where the motion starts at a minimum by adding a negative sign to the cosine function.

🎯 Recap of Oscillatory Motion Formula

The final section summarizes how to represent the motion of a simple harmonic oscillator using an equation based on amplitude, sine or cosine, and angular frequency. The equation can be fine-tuned to fit different scenarios, depending on whether the motion starts at a maximum, minimum, or equilibrium. By adjusting the period and amplitude, the function accurately represents the motion’s position at any given time. The recap includes guidance on when to use positive or negative sine and cosine functions.

Keywords

💡Simple Harmonic Oscillator

A simple harmonic oscillator refers to a system where a mass is displaced from its equilibrium position and then experiences a restoring force proportional to the displacement, leading to periodic motion. In the video, the instructor discusses how the motion of such an oscillator can be represented on a position graph, emphasizing the periodic nature of the system.

💡Amplitude

Amplitude is the maximum displacement of the oscillator from its equilibrium position. It represents the peak value on a position-time graph. The instructor explains that amplitude can vary, and this variation can be represented on the graph by stretching it vertically. For example, an amplitude of 0.2 meters is mentioned, representing how far the mass is displaced.

💡Period

The period (T) is the time it takes for one complete cycle of motion. It is the time between two consecutive peaks, troughs, or any corresponding points on the graph. The video explains that the period can be adjusted by stretching the graph horizontally, and it is crucial for determining the oscillatory behavior of the system.

💡Cosine Function

The cosine function is used to describe the motion of an oscillator that starts at its maximum displacement at time t=0. The instructor uses this function to model the position of the mass as a function of time, explaining that if the motion starts at a peak, a cosine function is appropriate.

💡Sine Function

The sine function is another trigonometric function used to describe oscillatory motion. It is typically used when the motion starts at the equilibrium position and moves upward. The video contrasts sine with cosine, showing that sine starts at zero, making it suitable for different initial conditions of the oscillator.

💡Angular Frequency (ω)

Angular frequency, often denoted as ω, is a measure of how quickly the oscillator cycles through its motion, and it is related to the period by the formula ω = 2π/T. The video explains that this term allows the function to adjust the period of the motion, ensuring the graph resets after the correct duration, matching the system's period.

💡Position-Time Graph

A position-time graph visually represents the oscillator's displacement over time, showing the cyclical nature of the motion. The instructor explains how different properties of the oscillator, like amplitude and period, influence the shape and features of this graph, making it a vital tool for analyzing harmonic motion.

💡Equilibrium Position

The equilibrium position is the point where the mass would naturally rest if there were no forces acting on it. It is the central point around which the oscillations occur. The video frequently references this as the baseline from which displacement, amplitude, and other measurements are taken.

💡Cyclic Process

A cyclic process refers to any motion or process that repeats itself in a regular pattern, such as the back-and-forth motion of a harmonic oscillator. The instructor illustrates how the motion of the mass can be represented on a unit circle to visualize this cyclic behavior.

💡Unit Circle

The unit circle is a concept from trigonometry that helps in understanding the cyclic nature of trigonometric functions like sine and cosine. In the video, the instructor uses the unit circle to explain how the position of the oscillator at any point in time can be mapped onto a circle, linking angular frequency to the oscillator's motion.

Highlights

Introduction to simple harmonic motion, represented on a horizontal position graph.

Explanation of amplitude as the maximum displacement from equilibrium.

Definition of the period, which is the time it takes for the motion to complete one full cycle.

You can stretch the graph vertically to change amplitude or horizontally to change the period.

Goal: find an equation that describes the horizontal position of a mass in motion as a function of time.

Choosing cosine for the equation because it starts at a maximum at T = 0.

Amplitude (A) is introduced into the equation to account for larger or smaller displacement.

Angular frequency (omega) is added to the equation to control the period of the oscillation.

The period is related to angular frequency by the formula omega = 2pi / T, where T is the period.

Graph represents a cyclic process that can also be modeled on a unit circle.

Cosine function can be modified by the amplitude and angular frequency to match any specific oscillator.

The resulting equation describes how the position of the mass changes with time.

Explanation of different function choices: cosine for starting at max, sine for starting at equilibrium.

Negative cosine is used if the motion starts at a minimum, allowing flexibility in the equation.

Recap of how to choose between sine and cosine functions based on the starting position of the motion.

Transcripts

- [Instructor] Alright, so we saw that you could represent

the motion of a simple harmonic oscillator

on a horizontal position graph and it looked kinda cool.

It looks something like this.

And the amplitude of that motion,

the maximum displacement from equilibrium

on this graph was just represented

by the maximum displacement from equilibrium,

it looked like this.

And the period, which was the time it took

for this entire process to reset, capital T,

is the period, is the time it takes to reset

was the time it takes to reset,

which would be from peak to peak

or from trough to trough

or from any point to any analogous point

on that cycle, this was the period T.

And so, with a graph that's a sine or a cosine,

you could represent any motion you want.

So, if you had some oscillator that had a larger amplitude,

you can imagine just stretching this thing vertically,

the period would stay the same,

but you could stretch out the amplitude.

Or, if you had something with a larger period,

you can imagine stretching it out horizontally

and leaving the amplitude the same,

or stretch it both ways to represent any oscillator

you want, which is kinda cool.

However, a lot of times you also need

the equation, in other words,

you might wanna know what equation would describe

this graph right here.

What equation would represent this graph here?

First of all, what do I even mean by like,

the equation for this graph?

What I mean is that this graph's representing

the horizontal position, X,

which is how far the mass has been displaced

this way from equilibrium, as a function of time.

So, we want a function that will be,

alright, what is the value of the position of this mass

as a function of time?

So, what would this equation be?

Gonna be a function, in other words,

you're gonna feed this function anytime you want,

and the function's gonna give you,

it's gonna spit out a value for the position,

and that should represent whatever position

this graph is representing, where the mass is at,

because the graph should agree

with what this function's gonna tell us.

And this function would tell us where the mass is

at any given moment.

So, what would this look like?

Now, we saw, like, this is a sine or a cosine, right?

So, this is either a sine or a cosine.

That's the first choice.

Do we wanna pick sine or cosine?

And what I always do, is I just look at the beginning

and I say, alright, in a T equals zero,

this one's starting at a maximum.

So, I wanna use cosine

because cosine starts at a maximum

and by starting at a maximum,

I mean, think about it, for a cosine of zero,

if you remember your trig functions,

cosine of zero is equal to one.

And so, because this is as big as cosine ever gets.

Sine and cosine can only ever get as big as one.

This thing's starting at a maximum.

So, cosine starts at a maximum at T equals zero.

This function here starts at a maximum at T equals zero.

I'm gonna wanna use cosine

but I'm gonna have to add a few elements in here.

Just cosine alone isn't gonna do it for me,

because cosine only gets as big as one.

This thing has to get as big as A,

whatever A is, this thing has to get that big.

So, in other words, my simple harmonic oscillators

aren't always gonna have an amplitude of one,

so I need some variable in here

that will represent what the amplitude is

for that given simple harmonic oscillator.

Let me make this less abstract.

Let me just say, let's say we happened

to pull this thing back 20 centimeters for .2 meters.

So, let's say our amplitude

for a particular simple harmonic oscillator

happened to be .2 meters,

that would mean that this here,

I can represent this here with .2 meters,

this doesn't even make it to one.

So, if I just left this as cosine,

that would say this thing's gonna get as big as one

at some point in time and that's a lie.

This thing only gets as big as .2, so it's easy though.

You might realize, if you clever,

we'll just multiply the front of this thing

by the amplitude, whatever the amplitude is multiplied,

'cause then one times amplitude

means that this X only gets as big as the amplitude,

which is exactly what I want.

I want this thing to be as big

as whatever the amplitude is of the motion.

And then there's one more piece,

you can't, you might be, like,

alright, we're done, I'm just gonna stick cosine of T

in here, that's not gonna work.

We do want this to be a function of time, right?

We wanna be able to plug in a time

and have this function spit out

what is the value of the position of the graph,

and that would represent where is it.

So, is it at .2, is it at .1, is it at .045,

or something like that?

That's what this function's supposed to do.

But just plugging in T here,

just having T alone, isn't gonna be good

because that would mean, look at,

a cosine of zero, we know cosine's one.

When does cosine get back to one?

That's gonna be when the inside,

the argument in here, is two pi.

So, we're gonna be using radians.

You could use degrees if you wanted to,

but most physicists and professors and teachers

are gonna be using radians for this case.

So, a cosine of two pi would again be one

because that's when, if you remember your unit circle,

that's when this function for cosine

has gone around one whole time

and it gets back where it started, right?

So, if something rotates through an angle of two pi,

you've reset the whole thing and that process has reset.

But that would mean this function resets

every two pi seconds, right?

'Cause at T equals zero, the function was one,

and then at T equals two pi, the function's one again.

That would mean the period for cosine of T

is two pi but our period isn't necessarily two pi, right?

Unless you got a really special case,

the period is whatever the period is.

Let's say it happened to be,

let's say our period happened to be like six seconds

for this particular case.

So, if this was six seconds,

we would not want a function

that resets after two pi seconds,

we need a function that resets after,

for this case, six seconds.

So, how do we do that?

Well, we have to not just have T in here.

We saw that if we just have T,

the period is always two pi,

'cause that's when cosine of T resets.

How would we do this?

Well, we're gonna be clever.

And if you're really clever you realize,

alright, I'm just gonna add a little variable in here.

I'm just gonna a little variable, boom, omega

and then multiply that by T,

and then I can tune this omega however I want, right?

If I can make omega big or small,

I can make the period of this function whatever I want.

And if you're curious, you might be like,

wait a minute, omega, we've used that before,

and you'd be right.

Omega we have used before.

That was the angular velocity

and remember, angular velocity was delta theta

over delta T, the amount of change in angle

over the amount of change in time,

which you might think isn't relevant here

'cause this mass is just going back and forth.

This mass isn't actually rotating in a circle.

However, you can represent repeating processes,

cyclic processes, processes that go through a cycle

on a unit circle.

So, in other words, let's say you start right here, right?

So, at T equals zero, you start,

we pulled this mass back and then we let go.

So, we start right there.

That would be right here on this unit circle

and then it flies through the equilibrium point,

that would be through a quarter of a cycle,

that means it would have made it to right here.

And then it makes its way over to this edge,

fully compresses this thing that would be over to here,

that would be through half a cycle,

and that would come back through,

let me find another dark color,

it would come back through the equilibrium point

and that would be down here.

And then we would get back to the initial point

and that would be one whole cycle.

So, you can see how we can represent cyclic processes

on a unit circle and that's how this makes sense.

That might seem abstract but it's really useful

'cause watch what we could do.

Naively you might think, alright,

how would we even define this?

Well, one cycle on a unit circle

is two pi radians, right?

If we're using radians, then one cycle would be two pi

'cause two pi is once around the circle.

And how long does that take?

Well, I know for a simple harmonic oscillator,

we defined the period to be the time it takes

for one whole cycle.

So, we'd have two pi over the period

and this is what you would plug in down here.

So, it turns out this does work.

So, even naively, just using our ideas of angular velocity,

plugging in two pi over the period,

will give us a function that resets

exactly when we want it to.

And you might not be convinced.

And if that doesn't make sense, I don't blame ya.

I might be confused too.

So, let me show you what I mean.

In other words, we take this function,

instead of writing omega, we can just do this.

We can just be like, alright, forget this,

taken this, omega is the angular velocity,

sometimes it's called the angular frequency,

in this case, so people use different terminologies.

You'll hear it as angular velocity or angular frequency.

If you take this angular velocity or angular frequency,

we just smack that right in here.

So, we just put that in there for omega,

and then multiply by T.

Watch what happens, this is beautiful.

So, if we take this, now it's gonna work.

So, we multiply by T.

T is our variable.

So, little t is our variable, two pi's the constant,

the period capital T is also a constant,

it'll be different for different harmonic oscillators.

But for a given harmonic oscillator,

capital T the period is a constant.

So, watch what happens now.

At T equals zero, this whole inside becomes zero.

So, let's say I plug in T equals zero.

We get to plug in little t whatever we want.

That is our variable, so if I plug in little t

equals zero, cosine of zero gives me one.

But now what happens?

If I plug in t equals, alright, after one whole process,

right, after one whole cycle,

it's gone through one whole period,

so if I plug in little t as capital T, the period.

Look what happens.

This capital T cancels with that capital T

and you just get two pi in here

and the cosine of two pi is also one.

That means this thing goes through a cycle

every capital T, period.

That's what we wanted.

We didn't want something that always had to have

two pi as the period.

Now we've got a function that we can plug in

whatever our period is down here.

That way, whenever this little t makes it to the period,

capital T, this whole argument in here becomes two pi

and the cosine resets itself

and you get a graph or a function that will give you a graph

that resets every period, which is exactly what we wanted.

So, in other words, to make this less abstract,

let's take this thing here,

for this particular function here,

for this particular choice of amplitude and period,

we could say that the graph that's representing this,

so the function that would represent this here,

instead of amplitude, we'd plug in .2.

So, 0.2, let me try to fit it in here,

0.2, I don't wanna put the units down here,

meters times cosine, remember, we wanted cosine

'cause it starts at a maximum

and this graph started at a maximum.

If it started down here and went up, I'd use sine

because sine starts at zero.

But this one started at a maximum.

And I have two pi over the period,

I can't just leave that as period T,

that's a little bit vague, I'd put in my actual period

and we said that the actual period

for this mass on a spring was six seconds.

And then little t, a lot of times people get confused,

they're like, alright, what do I plug in for little t?

You don't, typically, like, if you just want

the function for the position as a function of time,

you leave little t as the variable.

That's the variable that you have sitting here, right?

If I wanted to know what is the value

of the position of this mass at nine seconds,

I would plug in nine seconds.

I would calculate this function

with the nine seconds in there,

that would be the position at nine seconds.

Or, if I wanted the position at 12.25 seconds,

I'd plug in 12.25 seconds for our little t time,

calculate this function, plug it into the calculator

in other words and that would give me

the position at 12.25 seconds.

That's what this function can do for you.

That's how it can represent the motion

of a simple harmonic oscillator.

And now you might be like, dude, that took a long time.

Do they all take that long?

No, once you get good at this, it's really easy.

Watch, let me get rid of all that.

Let's say you got this problem on a test

or a quiz or whatever, on homework,

and it was like, hey, make an equation

that describes this simple harmonic oscillator.

It's easy.

First thing you do, do I want to use sine or cosine?

So, you might be like, oh, crud,

it doesn't start at a maximum

and it doesn't even start at zero, sine would start there.

It starts down here, but that's okay.

It starts at a minimum.

So, we're still gonna use cosine.

So, we're gonna say that X as a function of time

is gonna be, well, what's the amplitude?

The amplitude here is three meters.

So, three meters is our amplitude

because that's the maximum displacement

from equilibrium, so I'm gonna have

three meters out front

and then I'm gonna do cosine

because it starts at an extreme value,

like either a maximum or a minimum value.

Cosine of, and then I need two pi over the period.

What is my period?

I look at my graph and I ask,

how long does it take to reset?

So, started down here at a minimum,

when does it get back to a minimum?

That took four seconds.

So, four seconds would be the period,

so it'd be two pi over four seconds

and then little t, what do I plug in for little t?

I don't.

This is the variable that sits there

and waits for me to plug in whatever I want.

So, that's my variable little t that X is a function of.

But I'm not done.

This would be a graph that starts up here

and goes down like that.

This graph starts down here but that's easy.

You just multiply by a negative sine out front

and you've turned your cosine into negative cosine

and negative cosine starts down here.

So, note our amplitude is still three.

If the question asked, what is the amplitude?

The amplitude is the magnitude of the displacement,

maximum displacement, so that's still positive three meters,

even though it started down here,

but you could just include an extra negative out front

that essentially goes along with the cosine.

That would give you negative cosine

and there you have it, that would be your function.

So, keep in mind, it's good to remember,

if you start up here,

you're gonna wanna use cosine.

If you start down here,

you gonna wanna to use negative cosine.

If you start right here,

you're gonna wanna use sine.

If you start here and go up, that's gonna be sine.

And if you start here and go down,

that's gonna be negative sine.

That's what those functions look like.

So, recapping, you could use this equation

to represent the motion of a simple harmonic oscillator

which is always gonna be plus or minus

the amplitude, times either sine or cosine

of two pi over the period times the time.

This two pi over the period

is representing the angular frequency or angular velocity

and you would choose positive cosine

if you started at a max,

negative cosine if you started at a min.

Positive sine if you start at zero or equilibrium and go up.

Negative sine if you start at equilibrium and go down.