Pendulum Conservation of Energy

spsch
Hi, I have a general question to pendulums. I hope it is ok to post it in this format.
Please accept my apologies for my poor English.

Homework Statement

:
As a general Example:
I have a Pendulum of length L with Angle Theta as maximum displacement.

I know how to solve these problems. Find the period, the speed etc.
But,
I think I'm missing something fundamental here.
I would be very glad if you could help me understand.

So when the Pendulum is at maximum displacement it is also at maximum height in it's oscillation.
Which means it has the most potential energy in mgh.

So when it's in the middle shouldn't the total energy be 1/2Iw2 + 1/2mv2.

Why isn't mg(change in)h = 1/2Iw2 + 1/2mv2?

Am I missing some term?

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Hi, I have a general question to pendulums. I hope it is ok to post it in this format.
Please accept my apologies for my poor English.

Homework Statement

:
As a general Example:
I have a Pendulum of length L with Angle Theta as maximum displacement.

I know how to solve these problems. Find the period, the speed etc.
But,
I think I'm missing something fundamental here.
I would be very glad if you could help me understand.

So when the Pendulum is at maximum displacement it is also at maximum height in it's oscillation.
Which means it has the most potential energy in mgh.

So when it's in the middle shouldn't the total energy be 1/2Iw2 + 1/2mv2.

Why isn't mg(change in)h = 1/2Iw2 + 1/2mv2?

Am I missing some term?

What do you mean by ##\omega##? Is that the angular velocity about the fulcrum or the rotation of the bob about its centre of mass?

spsch
Hi, sorry, the angular velocity about the fulcrum.
What do you mean by ##\omega##? Is that the angular velocity about the fulcrum or the rotation of the bob about its centre of mass?

Apashanka
So when it's in the middle shouldn't the total energy be 1/2Iw2 + 1/2mv2.
If you consider the plane perpendicular to the string for θ=0,θ is the angle made by the string with the vertical.,as the reference for potential, at any angle θ the total energy is E=KE+mgl(1-cosθ),and for max. height E=mgl(1-cosθmax.) and for θ=0 ,E is simply KEmax.

spsch
To maybe add something, for the longest time I thought it should just be mgΔh = 1/2mv2 when the bob is in the middle.
I played around for a few hours and then come to 1/2Iω2 + 1/2mv2

now I realize that the bob is not rotating.

So to better formulate my question.

What is the total energy when theta is at it's max and what is it when the bob is in the middle at maximum velocity?

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Hi, sorry, the angular velocity about the fulcrum.

Is that not just an alternative way to express the kinetic energy? Try calculating ##I\omega^2## and ##mv^2## and see whether they are always the same for a pendulum bob.

spsch
If you consider the plane perpendicular to the string for θ=0,θ is the angle made by the string with the vertical.,as the reference for potential, at any angle θ the total energy is E=KE+mgl(1-cosθ),and for max. height E=mgl(1-cosθmax.) and for θ=0 ,E is simply KEmax.
Hi, thank you.

that is what I thought, but it doesn't work out numerically if you solve for v when θ=0.

spsch
Is that not just an alternative way to express the kinetic energy? Try calculating ##I\omega^2## and ##mv^2## and see whether they are always the same for a pendulum bob.
Hi, I don't think so. Consider a ball rolling down a ramp. mgh = 1/2Iω2 + 1/2mv2
the potential energy got converted to rotational and kinetic energy

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Hi, I don't think so. Consider a ball rolling down a ramp. mgh = 1/2Iω2 + 1/2mv2
the potential energy got converted to rotational and kinetic energy

I thought we were talking about a pendulum, not a ball rolling down a ramp.

Note that if you consider the pendulum bob as something greater than a point mass, then the kinetic energy of the pendulum becomes more complicated.

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Note that for a point mass, you have:

##I = ml^2## and ##v = \omega l##

Hence

##I\omega^2 = ml^2\omega^2 = mv^2##

And so the two expressions give the same KE of the point mass. Hence, you could use either. But not both added together.

Apashanka
Hi, thank you.

that is what I thought, but it doesn't work out numerically if you solve for v when θ=0.
v is max. for θ=0.,it has to be
Hi, I don't think so. Consider a ball rolling down a ramp. mgh = 1/2Iω2 + 1/2mv2
the potential energy got converted to rotational and kinetic energy
And an angle θ the tangential component of velocity is v=rθdot.=rω

And putting mv2=mr2ω2=Iω2,so they are the same.

PeroK
spsch
I thought we were talking about a pendulum, not a ball rolling down a ramp.

Note that if you consider the pendulum bob as something greater than a point mass, then the kinetic energy of the pendulum becomes more complicated.
Ah, I see what you meant. Yes, I misunderstood you. For the point mass, yes I is mr2 so Iω2 is mv2 since v = ωr.

But what I meant to ask. What is the total kinetic energy at the middle? is it not mgΔh?

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Ah, I see what you meant. Yes, I misunderstood you. For the point mass, yes I is mr2 so Iω2 is mv2 since v = ωr.

But what I meant to ask. What is the total kinetic energy at the middle? is it not mgΔh?

Yes, it must be.

Apashanka
Ah, I see what you meant. Yes, I misunderstood you. For the point mass, yes I is mr2 so Iω2 is mv2 since v = ωr.

But what I meant to ask. What is the total kinetic energy at the middle? is it not mgΔh?
Yes if Δh being the heighest displacement of the Bob from the reference potential level.

spsch
Hi, thank you both. I'm a slow typer. I'm sorry. I appreciate that for a point mass r2w2 turns out to be just v2

my first thought has been just mgΔh = 1/2mv2

then after a few hours and countless problems I thought I might be missing the rotational energy.
For neither I get the correct speed.
Which was wrong.

I realized that after Perok's first comment.

So my question is when theta is at max what is the total energy? is there anything else but mgh?

and when theta is zero what is the total energy is there anything else but 1/2mv2

spsch
Yes, it must be.
Thank you. So is there anything else but 1/2mv2 because if you solve for v I always get the wrong answer.

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Thank you. So is there anything else but 1/2mv2 because if you solve for v I always get the wrong answer.

You should post what you're doing. There is nothing more for a point mass. If the bob is larger, then it's not all traveling at the same speed. But, that's irrrelevant here.

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Thank you. So is there anything else but 1/2mv2 because if you solve for v I always get the wrong answer.

Just a thought. If you are using ##\omega = \dot{\theta}##, you may be confusing that the angular frequency of pendulum's motion - which is something different.

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Note that for a point mass, you have:

##I = ml^2## and ##v = \omega l##

Hence

##I\omega^2 = ml^2\omega^2 = mv^2##

And so the two expressions give the same KE of the point mass. Hence, you could use either. But not both added together.
Since they are equal, he can take half of each and add them to get the same thing. That is a strange way of doing it, but it will give him the correct answer every time.
CORRECTION: I forgot that the sum will be two times two great.

Last edited:
spsch
I'm sorry. Also for being so slow, I have real trouble with science notation on my laptop.

I have done about 30 problems yesterday and today and some a few days ago.
I get v usually by x = Lsin(theta)cos(wt)
so the derivative is v = Lsin(theta)ωsin(wt) hence v is Lsin(theta)ω (sin(90) = 1)

This would always give me the correct answer.

but mgΔh = 1/2mv2

squareroot(2gΔh) = v is never right. In every single problem.

Now I feel dumb and like I may be making algebra mistakes

Homework Helper
Gold Member
Hi, I have a general question to pendulums. I hope it is ok to post it in this format.
Please accept my apologies for my poor English.

Homework Statement

:
As a general Example:
I have a Pendulum of length L with Angle Theta as maximum displacement.

I know how to solve these problems. Find the period, the speed etc.
But,
I think I'm missing something fundamental here.
I would be very glad if you could help me understand.

So when the Pendulum is at maximum displacement it is also at maximum height in it's oscillation.
Which means it has the most potential energy in mgh.

So when it's in the middle shouldn't the total energy be 1/2Iw2 + 1/2mv2.

Why isn't mg(change in)h = 1/2Iw2 + 1/2mv2?

Am I missing some term?

The two haves are equal. ##L\omega^2 = mv^2## So you can write the expression in either term only or partition the total between them as you have.
CORRECTION: I forgot that the sum will be two times two great.

Last edited:
spsch
spsch
Since they are equal, he can take half of each and add them to get the same thing. That is a strange way of doing it, but it will give him the correct answer every time.
Thanks, I try to always check my answers by getting to them in two alternative ways. For motion problems I usually use conservation of energy for my alternative. Works fine in all problems except pendulums. So I feel I'm not seeing something.

Now i realize I maybe messing up algebra. I'll have to try again.

spsch
The two haves are equal. ##L\omega^2 = mv^2## So you can write the expression in either term only or partition the total between them as you have.
Hi, I got to this too at one of my last attempts but v was still wrong.

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I'm sorry. Also for being so slow, I have real trouble with science notation on my laptop.

I have done about 30 problems yesterday and today and some a few days ago.
I get v usually by x = Lsin(theta)cos(wt)
so the derivative is v = Lsin(theta)ωsin(wt) hence v is Lsin(theta)ω (sin(90) = 1)

This would always give me the correct answer.

but mgΔh = 1/2mv2

squareroot(2gΔh) = v is never right. In every single problem.

Now I feel dumb and like I may be making algebra mistakes

How are you calculating ##\Delta h##?

spsch
spsch
How are you calculating ##\Delta h##?
L minus cos(theta)L

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L minus cos(theta)L

Where are you getting ##\theta_{max}## from?

spsch
spsch
Where are you getting ##\theta_{max}## from?
Theta is usually given, it differs from problem to problem

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Theta is usually given, it differs from problem to problem

You need to post a problem. Show your calculations and we can see where you are going wrong. You seem to understand everything so I don't know what it could be!

spsch
spsch
You need to post a problem. Show your calculations and we can see where you are going wrong. You seem to understand everything so I don't know what it could be!
thanks. yeah it seems like I've been making some algebra mistakes. just seemed so unlikely by the amount of problems I've tried. But experience tells me there is no limit to dumb mistakes.

I'll try again tomorrow (I don't have the books with me here) and post if i continue to get it wrong.

Thank you very much, so sorry for being difficult!

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it seems like I've been making some algebra mistakes. just seemed so unlikely by the amount of problems I've tried.
That's exactly when they start happening again -- and in the parts that one is very confident of and doesn't think about enough.

spsch
spsch
You need to post a problem. Show your calculations and we can see where you are going wrong. You seem to understand everything so I don't know what it could be!
Hi, so I tried again. I think I'm using the calculator wrong.
When I use √2gΔh = v that still doesn't work out.
But if I use a correct value for v, square it and divide it by 2 I get gΔh. So mgΔh is 1/2mv2 after all. I'm so relieved. Sorry for the trouble. And thank you all for helping me!