Circular Motion, Velocity at top/bottom question

[swooshfactory]

Homework Statement

In an amusement park ride, passengers stand inside a 16-m-diameter roating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane.
Suppose the ring rotates once every 4.5 seconds. If a riders mass is 55 kg, with how much force does the ring push on her at the top of the ride? At the bottom?

Homework Equations

The Attempt at a Solution

First I solved for velocity:

8 m radius * 2pi = 16pi m circumference
16pi m / 4.5 s = 11.17 m/s

Then acceleration:
a=(v^2)/r = (124.77)/8= 15.6 m/s^2

Sum of forces in radial direction at top = n + mg = ma = 55 kg * 15.6 m/s^2 = 857.8 N
n + mg = 857.8 N


If I follow this logic, I would get the same answer for the sum of forces at the bottom.
My book gives the equation n=mg + [m(v_top)^2]/r and n= [m(v_bottom)^2] - mg .
I understand this, because at the top, the normal and gravitational force point in the same direction, and at the bottom they point in different directions. The problem is, I don't know how to computer the velocity at the top or at the bottom. My book also gives the equation v_c = sqrt(rg) which is the slowest speed at which the object can complete rotation, but I don't think this ride is traveling near that speed.

Can anyone tell me how to find the velocity at the top or bottom of circular motion? I'm pretty sure this is uniform, but I'm not sure.

 

[tiny-tim]

In an amusement park ride, passengers stand inside a 16-m-diameter roating ring. After the ring has acquired sufficient speed, it tilts into a vertical plane.
Suppose the ring rotates once every 4.5 seconds. If a riders mass is 55 kg, with how much force does the ring push on her at the top of the ride? At the bottom?
…
Can anyone tell me how to find the velocity at the top or bottom of circular motion? I'm pretty sure this is uniform, but I'm not sure.

Hi swooshfactory!

Yes it is uniform … the question makes that clear.

It's a powered ride.

 

[baba89]

To find the velocity at the top or bottom of circular motion, you can use the equation v = ωr, where v is the tangential velocity, ω is the angular velocity, and r is the radius. In this case, the radius is 8m and the angular velocity is 2π/4.5s, which gives us a tangential velocity of 11.17 m/s. This is the same velocity you calculated earlier.

In circular motion, the velocity is constant, so the velocity at the top and bottom of the ride will be the same. Therefore, the force at the top and bottom will also be the same, as you calculated earlier.

It is important to note that the equations given in your book are for a different scenario, where the velocity at the top and bottom may not be the same. In this scenario, the velocity is constant, so the equations are simplified.

I hope this helps clarify the concept of velocity in circular motion. Keep up the good work on your calculations!