Is the expression still valid for large differences in radius?

[AdityaDev]

If a ball rolls in a semicircular track starting from one end of track,( the track is kept vertical) and if radius of ball is r and radius of track is R
is this expression correct? (When ball reaches lowest point)
##mg(R-r)=\frac{1}{2}I_0\omega^2 + \frac{1} {2}I\omega_1^2+\frac{1}{2}mv^2##
where ##I_0## is the moment of inertia about centre of track since the ball moves in a circle and moment of inertia is mr^2
The second is for rotation of ball
the third is translational kinetic energy

what changes should I make if R is very very larger than r?

 

[Khashishi]

You are double counting the motion of the ball. The translational kinetic energy of the ball is the same thing as the rotational energy about the center of the track, so the I0 term is redundant.

 

[Doc Al]

Your first and third term (on the right hand side) represent the same energy. Once is enough.

(Khashishi beat me to it!)