Circular Motion and Radius of Curvature

[ThatMathGuy]

Homework Statement

Oh boy... Here we go...

At what minimum speed must a roller coaster be traveling when upside down at the top of a circle so that the passengers will not fall out? Assume a radius of curvature of 7.4 m.

I honestly have nothing. To begin with, I drew a free-body diagram but got nowhere considering the teacher never explained 'radius of curvature'. I've tried searching on the Internet but all I get are a bunch of formulas in geodesy and aren't applicable to the problem. It's not in the book either (Giancoli PHYSICS Updated Edition 2009).

Homework Equations

Ac=v^2/R
$$\SigmaF$$=mv^2/R

The Attempt at a Solution

Tried setting up a free-body diagram and got nowhere.

 

[Delphi51]

At the top of the circular motion, gravity must provide no more than the centripetal force needed to hold the car in circular motion. If Fg exceeds Fc, the car will fall off the track. So start with Fc = Fg, fill in the detailed formulas and solve for v.

 

[tomcue]

Also tried searching online and in the textbook but couldn't find anything relevant.

Dear student,

I understand your frustration with this problem. First of all, the term "radius of curvature" refers to the radius of the circle that the roller coaster is traveling on. In this problem, the radius of curvature is given as 7.4 m.

To solve this problem, we need to use the equation for centripetal acceleration, which is Ac = v^2/R, where v is the velocity and R is the radius of curvature. We also need to consider the force of gravity, which will act downwards on the passengers.

To prevent the passengers from falling out, the centripetal acceleration must be equal to or greater than the force of gravity. So, we can set up the following equation:

Ac ≥ mg

Where m is the mass of the passengers and g is the acceleration due to gravity.

We can substitute in the equation for centripetal acceleration, and rearrange to solve for the minimum velocity:

v ≥ √(Rg)

Plugging in the values for R and g, we get:

v ≥ √(7.4 * 9.8) = 8.58 m/s

So, the minimum speed that the roller coaster must be traveling at the top of the circle is 8.58 m/s.

I hope this helps clarify the problem for you. If you have any further questions, please don't hesitate to ask. Keep up the good work in your physics studies!

Best regards,