- #1

Jamie_Pi

- 18

- 0

## 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!