# Starting height of marble rolling around a loop the loop

• Jamie_Pi
In summary, the marble needs to have a minimum height of h to make it around the loop-the-loop without falling off.
Jamie_Pi

## Homework Statement

The marble rolls down a track and around a loop-the-loop of radius R. The marble has mass m and radius r. What minimum height h must the track have for the marble to make it around the loop-the-loop without falling off? (Use any variable or symbol stated above along with the following as necessary: g.)

## Homework Equations

KE=1/2mv^2+1/2I*w^2
KE+PE+W=KE+PE+∆Eth
mv^2/r=centripetal force
w=v/r

## The Attempt at a Solution

I started by imagining the marble was already at the bottom of the hill. What does the marble need to get all the way around the loop the loop? I set the kinetic energy of the ball at the bottom of the hill equal to the kinetic energy of the ball at the top of the loop plus the potential energy at the top of the loop:
KEi=KEf+PEf
Where KEi is the kinetic energy of the marble at the bottom of the hill and KEf is the kinetic energy at the top of the loop and PEf is the potential energy at the top of the loop. I imagined I could use this to solve for the potential energy at the top of the hill, since that should be equal to the kinetic energy at the bottom of the hill.
To solve for PEf, I used PE=mg*h, in this context, PEf=mg*2R (the height of the loop).

To solve for kinetic energy, I found the velocity required to loop the loop:
mv^2/R=mg (The centripetal force needs to exactly equal the force of gravity)
Therefore, v=sqrt(g*R).
I plugged this into KE=1/2mv^2+1/2Iw^2 (using the moment of inertia for a sphere) to get
KEf=1/2m*v^2+1/2*2/5mr^2*v^2/r^2 which becomes
KEf=1/2mgR+1/5mgR (plugging my value for v and simplifying)
So, I figure I'm done here. I put these back into my conservation of energy equation:
KEi=1/2mgR+1/5mgR+mg2R
And I think, "Well, I can assume that the marble fell from height h to get this kinetic energy, so all of the initial starting potential energy must have been converted into kinetic energy. Therefore, the starting potential energy must be equal to this kinetic energy."
And I write down:
PEi=KEf (what was once KEi is now KEf) which becomes
mgh=1/2mgR+1/5mgR+2mgR

h=R(1/2+1/5+2)

Which is wrong. I'm really appreciative that any of you read this whole thing, if you could point out any mistakes or wrong assumptions that I made, I'd be super thankful!

#### Attachments

• Screen Shot 2017-11-03 at 5.04.21 PM.png
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Jamie_Pi said:
PEf=mg*2R
Not quite. This is not a point particle.

Jamie_Pi
You almost have it, but you need to take into account the radius of the marble when finding potential energy.

Edit: I missed that this was already posted, didn't mean to bump with basically the same exact thing.

Jamie_Pi
haruspex said:
Not quite. This is not a point particle.
Ok, so instead of saying
mgh=R(1/2+1/5+2) I'll say
mg(h+r)=R(1/2+1/5+2) getting
h=R(1/2+1/5+2)-r

But that isn't right either, so I must be missing something. I can't find any other definition of gravitational potential energy online other than U=mgh.

Jamie_Pi said:
mg(h+r)
Still not quite there. Compare the initial position with the top-of-loop position.
You also need to consider how the centripetal acceleration is affected.

Jamie_Pi
haruspex said:
Still not quite there. Compare the initial position with the top-of-loop position.
You also need to consider how the centripetal acceleration is affected.

Ahhh, I think I understand:
KEi=1/2mg(R-r)+1/5mg(R-r)+2mg(R-r)
and
PE=mg(h+r)
meaning that
h=(R-r)(1/5+1/2+2)-r

Is that right?

Still not quite. You have not corrected this:
Jamie_Pi said:
PE=mg(h+r)

It might help to consider the extreme case, r=R.

Jamie_Pi
haruspex said:
It might help to consider the extreme case, r=R.
Is it
mg(h-r)?
I'm not really sure what to do other than that.

Jamie_Pi said:
Is it
mg(h-r)?
I'm not really sure what to do other than that.
Try drawing a diagram with h=2R. Compare the initial and final positions of the marble. How much has it dropped?

Jamie_Pi
haruspex said:
Try drawing a diagram with h=2R. Compare the initial and final positions of the marble. How much has it dropped?
Ok, I drew this: (sorry, poor quality)

And from the top of the hill, it seems that the marble falls as far as h is, which makes sense.
So is the potential energy at the top of the hill equal to mgh, but at the top of the loop equal to 2(R-r)? That would make the equation:
h=(R-r)(1/2+1/5+2)

#### Attachments

• Photo on 11-3-17 at 8.21 PM.jpg
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Jamie_Pi said:
Ok, I drew this: (sorry, poor quality)
View attachment 214299
And from the top of the hill, it seems that the marble falls as far as h is, which makes sense.
So is the potential energy at the top of the hill equal to mgh, but at the top of the loop equal to 2(R-r)? That would make the equation:
h=(R-r)(1/2+1/5+2)
Yes. It's the 2r drop, rather than r, that you were missing.

Jamie_Pi
haruspex said:
Yes. It's the 2r drop, rather than r, that you were missing.
Ok, I see. Thanks! You've been a big help.

Jamie_Pi said:
Ok, I see. Thanks! You've been a big help.
One interesting point about the answer... even if we let r tend to zero, the 1/5 term is still there. It is a common error to assume that if r<<R then the rotational energy can be ignored.

Jamie_Pi

## 1. What is the starting height required for a marble to successfully complete a loop the loop?

The starting height required for a marble to successfully complete a loop the loop depends on the size of the loop and the speed of the marble. However, generally a starting height of at least 5 times the diameter of the loop is recommended.

## 2. Can a marble complete a loop the loop without any initial height?

No, a marble requires a minimum starting height in order to have enough potential energy to complete a loop the loop. Without any initial height, the marble will not have enough speed to maintain its centripetal force and will fall off the loop.

## 3. How does the starting height affect the speed of the marble when completing a loop the loop?

The starting height affects the speed of the marble when completing a loop the loop as it determines the amount of potential energy the marble has. The higher the starting height, the greater the potential energy and the higher the speed of the marble will be when it reaches the loop.

## 4. Is there a limit to the starting height a marble can have when completing a loop the loop?

There is no specific limit to the starting height a marble can have when completing a loop the loop. However, a starting height that is too high can cause the marble to have too much potential energy and result in it flying off the loop. It is important to find the right balance between height and speed for a successful loop the loop.

## 5. How does the shape of the loop affect the required starting height for a marble?

The shape of the loop can affect the required starting height for a marble as different shapes will have different radii and therefore require different amounts of centripetal force. A wider, flatter loop will require a lower starting height compared to a narrower, steeper loop.

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