how many revolutions per minute would a 15 m ferris wheel need to make for passengers to feel weightless at the top?
To calculate the number of revolutions per minute for a 15 m Ferris wheel at which passengers would feel weightless at the top, we need to consider the gravitational force acting on the riders and the centripetal force that keeps them in circular motion.
At the top of the Ferris wheel, the riders would experience apparent weightlessness when the centrifugal force (centripetal force) acting outwards is equal to the gravitational force acting downwards.
First, let's calculate the gravitational force acting on the riders. The formula for calculating gravitational force is:
Fg = m * g
where Fg is the gravitational force, m is the mass of the rider, and g is the acceleration due to gravity (approximately equal to 9.8 m/s^2).
Next, let's calculate the centripetal force acting on the riders. The centripetal force is given by the formula:
Fc = m * (v^2 / r)
where Fc is the centripetal force, m is the mass of the rider, v is the linear velocity, and r is the radius of the Ferris wheel.
Since at the top of the Ferris wheel, the centripetal force should be equal to the gravitational force, we can set these two equations equal to each other:
m * (v^2 / r) = m * g
Now, we can solve for the linear velocity (v):
v^2 = r * g
v = sqrt(r * g)
Substituting the values r = 15 m and g = 9.8 m/s^2, we can calculate the linear velocity (v):
v = sqrt(15 * 9.8) = 14.37 m/s
Finally, to calculate the number of revolutions per minute, we need to convert the linear velocity into angular velocity. The angular velocity (ω) is given by the formula:
ω = v / r
Substituting the values v = 14.37 m/s and r = 15 m:
ω = 14.37 / 15 = 0.958 rad/s
To convert the angular velocity into revolutions per minute, we can use the conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds
So,
ω (in revolutions per minute) = (0.958 rad/s) * (1 revolution / 2π radians) * (60 seconds / 1 minute)
ω ≈ 5.77 revolutions per minute
Therefore, a 15 m Ferris wheel would need to make approximately 5.77 revolutions per minute for passengers to feel weightless at the top.
R w^2 would have to equal g.
w is the angular velocity in radians per second.
Solve for w and convert it to rpm