A roller coaster car moves on a track with one section that is a vertical circular loop of radius R. When the car is at the top of the loop it just maintains contact with the track. What is the car’s velocity at this point?

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BlackTom AI · Accepted Answer

Question restatement: A roller coaster car moves along a track that includes a vertical circular loop of radius R. At the top of the loop, the car just maintains contact with the track. What is the car’s velocity at this point?

Option 1: 0. This would imply no motion, which cannot be the case if the car is at the top of the loop and just maintaining contact; gravity alone would not provide a static condition for motion around a loop. Therefore this is incorrect.

Option 2: √(gR). This matches the standard condition for a mass in circular motion at the top of a vertical loop when the normal force goes to zero (just maintaining contact). At the top, the required centripetal acceleration is v^2/R directed downward. The only force providing this inward acceleration (with N = 0) is the weight mg acting downward, so m v^2 / R = m g, giving v^2 = gR and hence v = √(gR). This is the correct understanding of the contact condition.

Option 3: g/R. This has the wrong dimensional form for velocity. g has units of m/s^2 and R has units of m, so g/R would have units of 1/s^2 times sqrt(m) re-evaluations, not velocity. It misinterprets the relation between forces and motion required for circular motion; thus incorrect.

Option 4: Rg. This is simply g multiplied by R, which yields units of m^2/s^2, not velocity. It confuses the expression for v^2 with v, and does not satisfy the centripetal condition v^2/R = g when N = 0. Therefore incorrect.

Summary: The only option consistent with the physics of a mass undergoing circular motion at the top of a vertical loop with zero normal force is v = √(gR); the other choices fail due to incorrect dimensional analysis or incorrect physical reasoning about the forces and motion at the top.

SPHY001 Practice Quiz - Unit 06 | LA046

A roller coaster car moves on a track with one section that is a vertical circular loop of radius R. When the car is at the top of the loop it just maintains contact with the track. What is the car’s velocity at this point?

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