A roller-coaster car is released from rest from a height h and then moves freely

Question

A roller-coaster car 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 in terms of R.
(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 car's weight. The normal force on each rider follows the same rule.

Explanation / Answer

h = height of release R = radius of loop m = mass of roller coaster g = accel of gravity Coaster Total Energy at release point = PE = mgh {no KE} KE of coaster at TOP of loop = 1/2mV² {where V = tangential speed at top of loop} PE of coaster at TOP of loop = mg(2R) = 2mgR In order to experience the "weightless effect" at TOP of loop: Fc = mV²/R = mg cancel m's V²/R = g V² = Rg substitute in KE of coaster at TOP of loop: KE = 1/2m(Rg) Total Energy of coaster at TOP of loop = KE + PE = 1/2mgR + 2mgR = 2.5mgR Total Energy of coaster at release = mgh set Total Energies equal since no loss of energy thru friction mgh = 2.5mgR cancel "mg"s h = 2.5R

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.