As the chief design engineer for a major toy company, you are in charge of designing a loop-the-loop toy for youngsters. The idea is that a ball of mass m and radius r will roll down an inclined track and around the loop without slipping. The ball starts from rest at a height h above the tabletop that supports the whole track. The loop radius is R. Determine the minimum height h, in terms of R and r, for which the ball will remain in contact with the track during the whole of its loop-the-loop journey

h = 2.7 R - 1.7 r

Explanation:    

normally, force is 0 at the top ;

-N - mg = - m v^2 / ( R - r )

-0 - mg = (- mv^2) / ( R-r )

mg = (mv^2) / (R - r)

g = v^2 / ( R - r ) ; equation 1

conservation of energy ;

ΔK + ΔP = 0 ;

1/2 I ω^2 + 1/2 m v^2 + mg ( h2 - h1 ) = 0 ;  

0.5 * ( 2/5 ) m r^2 * ( v / r )^2 + 0.5 m v^2 + mg ( ( 2R - r ) -h ) = 0 ;

0.5 * ( 2/5 ) m r^2 * ( v / r )^2 + 0.5 m v^2 = mg ( - 2R + r + h ) ;

0.5 * ( 2/5 )r^2 * ( v / r )^2 + 0.5 v^2 = g ( - 2R + r + h ) ;

0.5 * ( 2/5 ) v^2 + 0.5 v^2 = g ( - 2R + r + h ) ;

[ 0.5 * ( 2/5 ) + 0.5 ] v^2 = g ( - 2R + r + h ) ;equation 2

from equation 1 ,  v^2 = g ( R - r ), input in equation 2

[ 0.5 * ( 2/5 ) + 0.5 ] [ g ( R - r ) ] = g ( - 2R + r + h )

[ 0.5 * ( 2/5 ) + 0.5 ] [ ( R - r ) ] = ( - 2R + r + h )

0.7 ( R - r ) = h - 2R + r

0.7R - 0.7r = h - 2R + r

solve for h

h = 0.7R + 2R - 0.7r - r

h = 2.7 R - 1.7 r

v=ir = 3x15 =45 v