At what minimum speed must a roller coaster be traveling when upside down at the top of a circle?

Question

At what minimum speed must a roller coaster be traveling when upside down at the top of a circle if the passengers are not to fall out. The radius of curvature is 7.54 metres. The mass of the passenger is 72.00 kg. [Answer in km/h]

A. 8.6 km/h
B. 30.9 km/h
C. 16.3 km/h
D. 13.3 km/h
E. 74.9 km/h

Answer 1

This is a force problem.

ID all the forces acting on the RC car (and you with mass m). First, there is weight, that's always there acting to pull you and the car into the center of the earth. Then there are centripetal (P) and centrifugal (C) forces, the car seat is pushing you inward along the radius of rotation (R) and your butt is pushing back on the RC car seat.

Now add them up as they work on you and the car. P and C are the same force magnitude, but one is the reaction to the other so they head in opposite directions. Thus, the net force acting on you with mass m is f = W - C = ma = 0 when your weight is exactly offset by the equal but opposite centrifugal force. f = 0 because you are not accelerating either into the center of the turn or out away from that turn. You are not falling out of the RC car.

Thus, we can write W = mg = mv^2/R = C; so that g = v^2/R when the two forces are equal and opposite. g is 9.81 m/sec^2 on Earth's surface and R = 7.54 m, the radius. So there you have it v = sqrt(g/R) and you can do the math.

But here's the physics...the answer does not depend on your m = 72 kg at all. Why? Because m in the W weight and m in the C force cancel out. In physics this often happens, factors will cancel out. This results because, in physics, we often are talking about equal and opposite physical characteristics.

Answer 2

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Answer 3

centripetal acceleration when moving along a circle is
a = v^2/R
you want a >= g
so
v^2 >= g*R
v >= sqrt(10*7.54)
== sqrt(75)
75 is between 64 and 81 os
== about 8.5 m/sec

Now, 10 m/sec is 36 km/h

so the answer is B (it is a little less that 36, just as 8 is a little less then 10 -- you do not need a calculator)

Answer 4

8.6 km/h