A roller-coaster car shown in the figure below is released from rest from a heig

Question

A roller-coaster car shown in the figure below is released from rest from a height h and then moves freely with negligible friction. The roller-coaster track includes a circular loop of radius R in a vertical plane (a) First suppose the car barely makes it around the loop; at the top of the loop, the riders are upside down and feel weightless. Find the required height h of the release point above the bottom of the loop. (Use any variable or symbol stated above along with the following as necessary: g.) (b) Now assume the release point is at or above the minimum required height. Show that the normal force on the car at the bottom of the loop exceeds the normal force at the top of the loop by six times the weight of the car. (Do this on paper. Your instructor may ask you to turn in this work.) The normal force on each rider follows the same rule. Such a large normal force is dangerous and very uncomfortable for the riders. Roller coasters are therefore not built with circular loops in vertical planes. The photograph shows an actual design.

Explanation / Answer

a) speed at top of loop = sqrt(2g(h-2R))

now mv2/R = mg

=> v2 = Rg => 2g(h-2R) = Rg => h- 2R = R/2 => h = 5R/2

b) speed at bottom = sqrt(2gh)

normal force on bottom = mv2/R + mg = 2mgh/R + mg

normal force at top = mv2/R - mg = 2mg(h-2R)/R - mg

= 2mgh/R - 4mg - mg = 2mgh/R - 5mg

=> bottom normal force - top normal force = 2mgh/R + mg - (2mgh/R - 5mg) = 6mg = 6 x weight of car

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