physics

A solid ball of mass m and radius r rolls without slipping through a loop of radius R, as shown in the figure. From what height h should the ball be launched in order to make it through the loop without falling off the track? (Use any variable or symbol stated above along with the following as necessary: g.)
h =

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  1. Ac at top of loop = g = v^2/R
    so
    v^2 at top of loop must be g R

    Height difference = (h-2R)

    so total Ke at top of loop = m g (h-2R)
    but
    total Ke = (1/2)m v^2 + (1/2)I w^2
    but v = w r so w = v/r

    total Ke = (1/2) [m + I/r^2]v^2
    but
    v^2 must be gR
    so
    total Ke = (1/2)[m + I/r^2]gR
    but we know total Ke is loss of Pe = m g (h-2R)
    so
    (h-2R) = (1/2)[ 1 + I/mr^2] R
    h = 2R + (1/2)[ 1 + I/mr^2] R
    I is (2/5) m r^2
    so
    h = 2R + (1/2) [ 7/5 ]R
    h = 2.7 R

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