Energy Conservation in Angular motion / Moment of Inertia

[EEristavi]

Homework Statement

A uniform ball of radius r rolls without slipping down from
the top of a sphere of radius R. Find the angular velocity of the ball
at the moment it breaks off the sphere. The initial velocity of the
ball is negligible.

Relevant Equations

K = I w^2 / 2
T = F R
U = m g h

I write Conservation of Energy:

Potential Energy loss(change):
U = m g $\Delta$h = m g (R+r) (1-cos$\alpha$)

kinetic Energy gain(change):
K = ($\frac {m v^2} 2$ + $\frac {I \omega^2} 2$) + ($\frac {M v_2^2} 2$ + $\frac {I_2 \omega_2^2} 2$)

U = K
m g (R+r) (1-cos$\alpha$) = ($\frac {m v^2} 2$ + $\frac {I \omega^2} 2$) + ($\frac {M v_2^2} 2$ + $\frac {I_2 \omega_2^2} 2$)

-----------------

However,
In the solution we have:

m g (R+r) (1-cos$\alpha$) = $\frac {m v^2} 2$ + $\frac {I \omega^2} 2$

-----------------

As I understand,
big ball doesn't roll (Or we don't consider it's rolling and movement).Need help here..

 

[kuruman]

Assume that the sphere of radius R is and stays at rest. At some point you need to write down the condition for separation of the rolling ball from the sphere.

 

[EEristavi]

At some point you need to write down the condition for separation of the rolling ball from the sphere.

Agreed, but first I need to understand what I asked above.

Assume that the sphere of radius R is and stays at rest

That's my question - why I must consider (or assume) that sphere of radius R stays at rest

 

[jbriggs444]

That's my question - why I must consider (or assume) that sphere of radius R stays at rest

There are no hard and fast rules for such. I would assume that it remains at rest because any other assumption would require that we have enough information to calculate the moment of inertia of the sphere whose radius is R and the slipperiness of the ground on which it rests.

Since we do not know those things and since the question is supposed to be answerable, they must not matter. The simplest explanation for why they do not matter is that the sphere whose radius is R remains at rest.

I would expect that it never dawned on the person setting the question that someone might worry whether the sphere underneath could move.

 

[kuruman]

That's my question - why I must consider (or assume) that sphere of radius R stays at rest

As @jbriggs444 already indicated, you need to make enough assumptions to be able to solve the problem with what is given. It also helps to have encountered a problem and its variants enough times so that you can understand the author's intentions. The standard formulation of this kind of problem has the mass either sliding without friction or rolling without slipping on a hemisphere that is firmly attached to the Earth.

 

[EEristavi]

Ok, Thank you.
I expected more from the Author... :D

 

[haruspex]

Ok, Thank you.
I expected more from the Author... :D

If you care to try the case where the large ball can move I'm sure you will get help from this forum. But note that it will make the centripetal acceleration tricky.