A circular-motion addict of mass 83.0 kg rides a Ferris wheel around in a vertical circle of radius 11.0 m at a constant speed of 6.10 m/s. (a) What is the period of the motion? What is the magnitude of the normal force on the addict from the seat when both go through (b) the highest point of the circular path and (c) the lowest point?
centripetal force is ... m v^2 / r
a) (2 π r) / 6.10 m/s
b) m [g - (v^2 / r)]
c) m [g + (v^2 / r)]
To find the period of the motion, we can use the formula:
T = 2πr/v,
where T is the period, r is the radius, and v is the velocity.
In this case, the radius of the circular path is 11.0 m and the velocity is 6.10 m/s.
Substituting these values into the formula, we get:
T = 2π(11.0 m)/(6.10 m/s)
T = 6.819 s.
So, the period of the motion is approximately 6.819 seconds.
To find the magnitude of the normal force on the addict from the seat, we can consider the forces acting on the addict at the highest and lowest points of the circular path.
(b) At the highest point, the only forces acting on the addict are gravity (mg) and the normal force (N). Since the addict is at the top of the circular path, there is no net acceleration in the vertical direction. Therefore, the normal force is equal to the gravitational force:
N = mg
Substituting the mass given as 83.0 kg, and the acceleration due to gravity as approximately 9.8 m/s^2, we have:
N = (83.0 kg)(9.8 m/s^2)
N = 813.4 N.
So the magnitude of the normal force on the addict from the seat at the highest point is approximately 813.4 N.
(c) At the lowest point, the forces acting on the addict are gravity (mg) and the normal force (N). The normal force provides the necessary centripetal force to keep the addict moving in a circular path. Therefore, the normal force is greater than the gravitational force.
The net force acting on the addict at the lowest point is the difference between the normal force and the gravitational force:
net force = N - mg
The net force is directed towards the center of the circular path.
Since we know that the centripetal force is mv^2/r, where m is the mass, v is the velocity, and r is the radius, we can equate the net force to the centripetal force:
N - mg = mv^2/r
Substituting the given values, we have:
N - (83.0 kg)(9.8 m/s^2) = (83.0 kg)(6.10 m/s)^2/(11.0 m)
Solving this equation, we find:
N = 1001.1 N.
So the magnitude of the normal force on the addict from the seat at the lowest point is approximately 1001.1 N.