Coin-vertical displacement question and other

[Elo21]

Coin--vertical displacement question and other

Homework Statement

A Perfectly hemispherical dome with a 10. meter radius is treated with a frictionless coating. A marble and a coin are released from the top of the dome simultaneously. We will assume there is no air friction. The coin will slide, not roll. The coin will hit the ground first because it has more kinetic energy.



2. Questions
Show mathematically whether or not the coin will stay in contact with the dome until it reaches the ground. If it does not, what will the coin's vertical displacement be at the instant it loses contact with the dome, and how far from the base of the dome will the coin hit the ground? (be clear and include explanations where necessary)

3. Attempt at the solution


This part is where I am confused.
I know that when the normal force is greater then the weight (or gravitational force) the coin will lose contact with the dome.

I am lost on how to show it mathematically and where to start.

Please Help!

 

[tiny-tim]

hello hello hello hello hello hello hello hello hello hello hello hello hello hello hello hello hello hello hello hello hello Elo21 !

know that when the normal force is greater then the weight (or gravitational force) the coin will lose contact with the dome.

Nooo …

the coin will lose contact when the normal force is zero.

 

[Elo21]

Okay.

So to solve it I would have to find where on the surface Fn=0, like the height from the radius/ diameter of the circle. Then using that use an equation to find the angle that it comes off at?

Sorry if I do not make sense. :(

 

[tiny-tim]

Yes, find the height or the angle where F_N = 0.

 

[Elo21]

so then...

if i go about finding the height using energy...

Total E= mgh+1/2mv²

and GPE equals mgh but because of the radius I can assume it equals mgr

GPE= total E= mgr

mgr=mgh+ 1/2mv²

Cancel m so I have

gr=gh+1/2v²

I know all variables now except for h and v and I want to solve for v.
So what would I put in for v to find h?

Once I know that I can do the rest... I just can't figure it out!

THANK YOU!

Sorry if that did not make sense...I am bad at explain things. :)

 

[tiny-tim]

(just got up :zzz: …)

so then...

if i go about finding the height using energy...

Total E= mgh+1/2mv²

and GPE equals mgh but because of the radius I can assume it equals mgr

No, that would mean h is constant.

You need v as a function of θ, so use h = r(1 - cosθ).

Try again. ​