Period dependence for mass on spring | Physics | Khan Academy

Summary

This educational video script explores the principles of a mass oscillating on a spring, focusing on amplitude and period. It clarifies that amplitude is determined by the displacement of the mass but does not affect the period. The period, instead, is solely dependent on the mass and the spring constant, as described by the formula T = 2π √(m/k). The script debunks the misconception that a larger amplitude leads to a longer period, explaining that the increased distance is compensated by a higher velocity due to Hooke's law. It also emphasizes that the period is independent of gravitational force, making the formula applicable in any gravitational environment.

Takeaways

• 🔍 The amplitude of oscillation is determined by the displacement of the mass from its equilibrium position.
• ⏱ The period of oscillation is the time it takes for the mass to complete one full cycle of motion.
• 🔄 Increasing the amplitude does not affect the period of oscillation; the mass travels farther but also moves faster, offsetting each other.
• 📐 Hooke's law states that the force exerted by a spring is proportional to the displacement from the equilibrium position.
• 📉 The period of a mass on a spring is given by the formula T = 2π √(m/k), where T is the period, m is the mass, and k is the spring constant.
• 🚀 The period is independent of the amplitude of oscillation, but it is affected by the mass and the spring constant.
• 🌌 The period of oscillation is not affected by gravity, so it remains the same even if the mass is oscillating vertically or horizontally.
• 📈 Increasing the mass results in a longer period due to increased inertia.
• 📉 Increasing the spring constant k results in a shorter period because the spring can exert a larger force, moving the mass more quickly.
• 🔍 The formula for the period of a mass on a spring can be derived using calculus and is applicable to both simple harmonic motion and energy considerations.

Q & A

What is the amplitude of an oscillating mass on a spring?

The amplitude is the maximum displacement from equilibrium, determined by the person or force pulling the mass back.

What is the period of oscillation for a mass on a spring?

The period is the time it takes for the mass to complete one full cycle of oscillation.

Does increasing the amplitude of oscillation affect the period?

No, changes in amplitude do not affect the period of oscillation. The period remains constant regardless of the amplitude.

What factors determine the period of a mass oscillating on a spring?

The period depends on the mass of the object and the spring constant, not on the amplitude of oscillation.

What is the formula for the period of a mass on a spring?

The formula for the period (T) of a mass on a spring is T = 2π √(m/k), where m is the mass and k is the spring constant.

Why does increasing the mass result in an increased period?

Increasing the mass increases the period because a larger mass has more inertia, making it more difficult to change its motion, thus taking longer to complete a cycle.

How does the spring constant affect the period of oscillation?

An increase in the spring constant (k) results in a smaller period because the force exerted by the spring is greater, allowing the mass to move more quickly through its cycle.

Does the direction of oscillation (horizontal or vertical) affect the period?

No, the direction of oscillation does not affect the period. The same formula applies to both horizontal and vertical oscillations.

Is the gravitational acceleration a factor in determining the period of a mass on a spring?

No, the gravitational acceleration does not affect the period of a mass on a spring. The period is independent of the gravitational constant.

Can the formula for the period of a mass on a spring be derived without calculus?

The formula can typically be derived using calculus, specifically in the context of simple harmonic motion. The instructor suggests watching videos on simple harmonic motion with calculus for the derivation.

Outlines

00:00

🔄 Understanding Amplitude and Period in Oscillating Mass Systems

In this section, the instructor explains the concepts of amplitude and period for a mass oscillating on a spring. The amplitude is defined as the maximum displacement from equilibrium and depends on how far the mass is pulled back. The period is the time it takes for one full cycle of motion. A key focus is whether the amplitude affects the period, with a thought experiment revealing that changes in amplitude do not alter the period. Although increasing amplitude means the mass travels a greater distance, the resulting faster speed perfectly offsets the increased distance, keeping the period constant. This is a crucial point to remember: amplitude changes do not affect the period.

05:01

🔍 Exploring the Factors That Affect the Period

This part explores the factors influencing the period of a mass-spring system. The formula for the period is given as \(T = 2\pi\sqrt{\frac{m}{k}}\), where \(m\) is the mass and \(k\) is the spring constant. Increasing the mass increases the period due to the mass's greater inertia, making it harder to move quickly. In contrast, increasing the spring constant decreases the period, as a stronger spring exerts a greater force, moving the mass faster through its cycle. The key takeaway is that the period depends on both the mass and spring constant but not on amplitude or gravitational forces. The formula applies to both horizontal and vertical oscillations, unaffected by gravity.

Keywords

💡Amplitude

Amplitude refers to the maximum displacement from equilibrium in an oscillating system, such as a mass on a spring. In the video, it is explained that the amplitude is determined by the force or action applied to displace the mass, and it is independent of the period. The video uses the example of pulling the mass back farther to create a larger amplitude, but clarifies that this does not affect the period of oscillation.

💡Period

The period is the time it takes for an oscillating system to complete one full cycle. The video emphasizes that the period is not affected by the amplitude of the oscillation, a counterintuitive concept. It is determined by other factors, such as the mass and the spring constant, as described by the formula provided in the video.

💡Equilibrium

Equilibrium in this context is the state of rest or balance where no net force acts on the oscillating mass. The video mentions that the amplitude is the maximum displacement from this equilibrium position. Understanding equilibrium is crucial for grasping the dynamics of the oscillating system.

💡Hooke's Law

Hooke's Law states that the force exerted by a spring is proportional to the displacement from its equilibrium position. The video uses Hooke's Law to explain why increasing the amplitude leads to a larger force from the spring, which in turn affects the velocity of the mass as it moves towards the equilibrium position.

💡Spring Constant

The spring constant (k) is a measure of the stiffness of a spring, as described by Hooke's Law. The video explains that the period of oscillation is inversely related to the spring constant; a larger spring constant results in a smaller period because the spring can exert a greater force, allowing the mass to move more quickly through its cycle.

💡Inertia

Inertia is the resistance of any physical object to a change in its state of motion or rest. The video relates inertia to the mass of the oscillating system, stating that a larger mass has more inertia, making it more difficult to change its motion and thus resulting in a longer period.

💡Velocity

Velocity is the speed of an object in a specific direction. In the context of the video, the velocity of the mass is influenced by the force exerted by the spring. The video explains that a larger force results in a higher velocity, which is one of the factors that offset the increased distance traveled at higher amplitudes.

💡Simple Harmonic Motion

Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. The video discusses the period of a mass on a spring as an example of simple harmonic motion, and mentions that the formula for the period can be derived using calculus.

💡Derivation

A derivation in mathematics refers to the process of obtaining a formula or a principle from a set of assumptions or previously established formulas. The video mentions that the formula for the period of a mass on a spring is derived using calculus, but does not provide the derivation within the script itself.

💡Gravitational Acceleration

Gravitational acceleration, often denoted as 'g', is the acceleration that an object experiences due to the force of gravity. The video clarifies that the period of oscillation for a mass on a spring does not depend on gravitational acceleration, which means the period remains the same whether the mass is oscillating on Earth, the moon, or Mars.

Highlights

The amplitude of oscillation is determined by the displacement from equilibrium.

The period of oscillation is the time for one complete cycle.

Amplitude is influenced by the force applied to displace the mass.

The period's dependency is less obvious and not immediately clear.

An increase in amplitude does not affect the period due to offsetting effects.

Hooke's law states that force is proportional to the amount the spring is stretched.

Greater amplitude results in a larger force from the spring, leading to higher velocity.

The period of a mass on a spring is independent of amplitude.

The period is determined by the formula \( T = 2\pi \sqrt{\frac{m}{k}} \), where \( T \) is the period, \( m \) is the mass, and \( k \) is the spring constant.

The period increases with mass due to increased inertia.

A larger spring constant \( k \) results in a smaller period due to a stronger force.

The formula for the period does not include gravitational acceleration, making it applicable in any gravitational field.

The period is the same for both horizontal and vertical oscillations of a mass on a spring.

The period is influenced by the mass and spring constant, but not by the amplitude or gravitational acceleration.

The formula for the period of a mass on a spring is derived using calculus and is applicable in various contexts.

Transcripts

- [Instructor] So, we saw that for a mass

oscillating on a spring,

there's a certain amplitude

and that's the maximum displacement from equilibrium.

But there's also a certain period,

and that's the time it takes for this process to reset.

In other words, the time it takes

for this mass to go through an entire cycle.

But what do these things depend on?

We know the definitions of them,

but what do they depend on?

Well, for the amplitude, it's kind of obvious,

the person pulling the mass back.

Whoever or whatever is displacing this mass

is the thing determining the amplitude.

So if you pull the mass back far,

you've given this oscillator a large amplitude,

and if you only pull it back a little bit,

you've given it a small amplitude.

But it's a little less obvious in terms of the period.

What does the period depend on?

Who or what determines 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.

It's pretty cool.

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?

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.

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.

That's why it increases the period.

That's why it takes longer.

So increasing the period means it takes longer

for this thing to go through a cycle,

and that makes sense in terms of the mass.

How about this k value?

That should make sense too.

If we increase the k value, look it,

increasing the k would give us more spring force

for the same amount of stretch.

So, if we increase the k value,

this force from the spring is gonna be bigger,

so it can pull harder and push harder on this mass.

And so, if you exert a larger force on a mass,

you can move it around more quickly,

and so, larger force means you can make this mass

go through a cycle more quickly

and that's why increasing this k gives you a smaller period

because if you can whip this mass around more quickly,

it takes less time for it to go through a cycle

and the period's gonna be less.

That confuses people sometimes,

taking more time means it's gonna have a larger period.

Sometimes, people think

if this mass gets moved around faster,

you should have a bigger period,

but that's the opposite.

If you move this mass around faster,

it's gonna take less time to move around,

and the period is gonna decrease

if you increase that k value.

So this is what the period of a mass on a spring depends on.

Note, it does not depend on amplitude.

So this is important.

No amplitude up here.

Change the amplitude, doesn't matter.

Those effects offset.

It only depends on the mass and the spring constant.

Again, I didn't derive this.

If you're curious, watch those videos that do derive it

where we use calculus to show this.

Something else that's important to note,

this equation works even if the mass is hanging vertically.

So, if you have this mass hanging from the ceiling,

right, something like this,

and this mass oscillates vertically up and down,

this equation would still give you the period

of a mass on a spring.

You'd plug in the mass that you had on the spring here.

You'd plug in the spring constant of the spring there.

This would still give you the period

of the mass on a spring.

In other words, it does not depend

on the gravitational constant,

so little g doesn't show up in here.

Little g would cause this thing to hang downward

at a lower equilibrium point,

but it does not affect the period of this mass on a spring,

which is good news.

This formula works for horizontal masses,

works for vertical masses,

gives you the period in both cases.

So, recapping, the period of a mass on a spring

does not depend on the amplitude.

You can change the amplitude,

but it will not affect how long it takes this mass

to go through a whole cycle.

And that's true for horizontal masses on a spring

and vertical masses on a spring.

The period also does not depend on

the gravitational acceleration,

so if you took this mass on a spring to Mars or the moon,

hung it vertically, let it oscillate,

if it's the same mass and the same spring,

it would have the same period.

It doesn't depend on what the acceleration due to gravity is

but the period is affected by the mass on a spring.

Bigger mass means you would get more period

because there's more inertia,

and it's also affected by the spring constant.

Bigger spring constant means you'd have less period

because the force from the spring would be larger.