R = radius of the loop
m = mass of the roller coaster car
for looping the loop critical speed at the top is given as
v₀ = critical speed at the top = sqrt(2gR)
given that :
v = speed of crossing at the top = 2 v₀ = 2 sqrt(2gR)
F = normal force at the top in downward direction
W = weight in downward direction = mg
force equation at the top is given as
F + W = m v²/R
F + mg = m (2 sqrt(2gR))²/R
F + mg = m (4 x 2 gR)/R
F + mg = m (8gR)/R
F + mg = 8 mg
F = 7 mg
F/(mg) = 7
so the ratio comes out to be 7:1
