A roller coaster rolls down an incline and into a vertical loop. The radius of t

Question

A roller coaster rolls down an incline and into a vertical loop. The radius of the loop on the track is R. Find the minimum height, h, of the incline for the roller coaster to be able to complete the loop without falling off the track. Ignore any types of friction. Give your answer in terms of R. Draw a free-body diagram of the mass at the top of the loop.

Explanation / Answer

See let the height of incline be H so the kinetic energy at the bottom of incline = mgH Now for completing the rotation around the circular track of radius R the roller coaster must first go a height 2R with a velocity u on the top such that the centripetal force due to the velocity u balances the weight of the object mu^2/R = mg => u^2 = Rg Now applying the energy conservation from the bottom of the track to top we have At bottom energy = mgH At top energy = mg2R +.5mu^2 =>ngH =mg2R +.5mu^2 => gH =2gR +.5gR => H = 2.5R

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