Show that on a roller coaster with a circular vertical loop the difference in your apparent weight at the top?

Show that on a roller coaster with a circular vertical loop, the difference in your apparent weight at the top of the loop and the bottom of the loop is 6 g's - that is, six times your weight. Ignore friction. Show also that as long as your speed is above the minimum needed, this answer doesn't depend on the size of the loop or how fast you got through it. Beginning height of the coaster is,h, and radius of loop is R.

Show that on a roller coaster with a circular vertical loop the difference in your apparent weight at the top?
Acceleration at top


A1=v1^2/r - g


Acceleration bottom


A2=v2^2/r + g





A2 - A1 = (v2^2 - v1^2)/r +2g





Energy at top = Et= KE + PE


=mv1^2/2 + mgh


h=2r


therefore


Et= mv1^2/2 + mg2r





Ebottom=Eb=mv2^2/2


but Eb=Et, therefore


Eb=mv2^2=mv1^2/2 + mg2r


therefore


(mv2^2 -mv1^2)/2 = mg2r


v2^2 -v1^2 = 4*gr


(v2^2 -v1^2)/r = 4*g





Now remember our equation from above





A2 - A1 = (v2^2 - v1^2)/r +2g





sub in (v2^2 -v1^2)/r = 4*g





A2 - A1 = 4g +2g


=6g



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