One popular ride at carnivals is the Roundabout, which works in the following manner. Riders stand on the inside edge of a circle, with their backs against a wall mounted on the edge of the circle. The wheel begins to spin. Once it is spinning fast enough the whole wheel tilts until the plane of the wheel is vertical (see picture). If the radius of the wheel is 8 meters, what angular velocity must the wheel have in rad/s so people don't fall off the wheel?

Question

One popular ride at carnivals is the Roundabout, which works in the following manner. Riders stand on the inside edge of a circle, with their backs against a wall mounted on the edge of the circle. The wheel begins to spin. Once it is spinning fast enough the whole wheel tilts until the plane of the wheel is vertical (see picture). If the radius of the wheel is 8 meters, what angular velocity must the wheel have in rad/s so people don't fall off the wheel?

Answer 1

Let ω=angular velocity (rad/s)

To offset gravity, the centripetal force must offset gravity, g, or
rω²=g

Solve for ω.

Answer 2

To determine the angular velocity required for people to not fall off the wheel, we need to consider the force acting on the riders due to the circular motion.

When the wheel is tilted and spinning, riders will experience a centrifugal force pushing them outwards. This centrifugal force must be balanced by the force of gravity acting towards the center of the wheel.

To calculate the angular velocity required, we can equate the centrifugal force and gravitational force acting on each rider.

The centrifugal force experienced by a rider can be calculated using the formula:

Centrifugal force = Mass of the rider × Radial acceleration

The radial acceleration is given by:

Radial acceleration = Radius of the wheel × Angular velocity²

The gravitational force acting on a rider can be calculated using the formula:

Gravitational force = Mass of the rider × g (acceleration due to gravity)

To keep the riders from falling off, the centrifugal force must be equal to the gravitational force. Therefore:

Mass of the rider × Radial acceleration = Mass of the rider × g

Since the mass of the rider is present on both sides of the equation, we can cancel it out:

Radial acceleration = g

Substituting in the expression for radial acceleration:

Radius of the wheel × Angular velocity² = g

Solving for angular velocity:

Angular velocity = √(g / Radius of the wheel)

Now we can plug in the values to calculate the angular velocity in radians per second:

Radius of the wheel = 8 meters
Acceleration due to gravity (g) = 9.8 m/s²

Angular velocity = √(9.8 / 8) ≈ √1.225 ≈ 1.105 rad/s

Therefore, the wheel must have an angular velocity of approximately 1.105 rad/s to prevent riders from falling off.

Let ω=angular velocity (rad/s)

1.106

To determine the angular velocity required for people to not fall off the wheel, we need to consider the force acting on the riders due to the circular motion.