What is the Solution to the Pendulum Energy Problem?

[PhizKid]

Homework Statement

Homework Equations

Mechanical conservation

The Attempt at a Solution

It says the energy at height d = (1/2)mv^2 + 2mg(L-d)
Isn't it just mgL = mgd + (1/2)mv^2 ? Since mgd is the final potential energy, at height d above the reference point which is taken to be the lowest point.

 

[BruceW]

It says the energy at height d = (1/2)mv^2 + 2mg(L-d)
Isn't it just mgL = mgd + (1/2)mv^2 ? Since mgd is the final potential energy, at height d above the reference point which is taken to be the lowest point.

It says that is the energy at half a revolution. So what height will the mass be above the reference point when it has gone through half a revolution? (the answer is not d).

 

[PhizKid]

Why isn't half a revolution d? Isn't it from the bottom of the peg to the top of the peg, which is halfway around the peg from the starting point? (If we take the vertical line down from the peg to the floor to be the reference point)

 

[grzz]

Let us take the lowest level of the bob (of mass m) as our reference level when any height of the bob is measured.

What is the height of the bob of mass m after half of a revolution?

 

[grzz]

As Bruce W said, this the height asked for in my previous post is not d.

 

[BruceW]

Isn't it from the bottom of the peg to the top of the peg, which is halfway around the peg from the starting point?

Do you mean the height is the distance from the position of the mass when it is at its lowest, to when it is at its highest? If that is what you mean, then yes I agree. But this distance is not d. Think about taking an actual piece of string, and what will the distance be? Maybe first think about what is the distance from the peg to the mass (hint: look at the diagram).

 

[PhizKid]

Then it's d/2

 

[grzz]

Try to find the radius with which the mass performs the half circle round the peg.

 

[BruceW]

Then it's d/2

You are not meant to assume that L is twice the length of d. (Although it might look like that from the picture).

 

[PhizKid]

The radius is d/2, then. Since the diameter is d. I don't understand the length we are trying to find. Half a revolution means 180 degrees, right? So we are looking for the subtended distance or the angular displacement?

 

[BruceW]

In the picture, what is the distance from the peg to the mass?

 

[PhizKid]

From the picture, it looks like peg to the mass is d/2. From the mass on the bottom to the dotted lines area at where d is. And the peg looks to be in the center of that.

 

[BruceW]

That's the problem. Peg to the mass is not d/2
Pivot to peg is d and the entire length of the string is L. So what is the distance from peg to the mass?

Edit: just to clarify, we are talking about the part of the picture where the mass is hanging straight down.

 

[PhizKid]

Ohhh. L - d then, lol

I see now...I thought the picture was saying that 'd' was marking the height from the very bottom of the picture..

 

[BruceW]

haha, I see what you mean now. Yeah, that is a bit confusing. Well, hopefully now you can see where their answer comes from.

Edit: to confirm, yes peg to mass is L-d