You are tasked with designing a roller coaster ride with a radius of curvature of 6.0 m. What minimum speed must the roller coaster have at the top of the circle so that the passengers experience a sense of "'free fall"?
g = v^2/R
To determine the minimum speed at the top of the roller coaster's circle for passengers to experience a sense of free fall, we can use the concept of centripetal force.
The force that keeps an object moving in a circular path is called centripetal force. In this case, the centripetal force is provided by the gravitational force acting on the passengers. At the top of the circle, this centripetal force is equal to the gravitational force acting on the passengers:
Fc = mg
where Fc is the centripetal force, m is the mass of the passengers, and g is the acceleration due to gravity (9.8 m/s^2).
The centripetal force can also be expressed in terms of the speed (v) of the roller coaster and the radius (r) of the circle:
Fc = mv^2 / r
Equating the two expressions for the centripetal force:
mg = mv^2 / r
Canceling the mass:
g = v^2 / r
Solving for v:
v^2 = gr
v = √(gr)
Substituting the given values:
v = √(9.8 m/s^2 * 6.0 m)
v ≈ √(58.8)
v ≈ 7.67 m/s
Therefore, the minimum speed the roller coaster must have at the top of the circle for passengers to experience a sense of free fall is approximately 7.67 m/s.