A Ferris wheel, 24.0 m in diameter, rotates once every 12.5 s(see the diagram.) What is the fractional change in a person'sapparent weight (a) at the top, and (b) at the bottom, as comparedto her weight at rest?
The net centripetal force is mr omega^2 where r is 24 and omega is 2pi/12.5 (the m's will cancel)
So fractional weight at the top is
(mg - mr omega^2) / mg and at the bottom
(mg + mr omega^2) / mg
And you can now see the m's do indeed cancel.
To calculate the fractional change in a person's apparent weight at the top and bottom of the Ferris wheel, we need to consider the centripetal acceleration experienced by the person.
(a) At the top of the Ferris wheel, the person is in the highest position. The apparent weight experienced by the person will be the sum of the gravitational force and the centripetal force acting towards the center. In this case, the gravitational force is acting downwards, while the centripetal force is acting upwards.
Since the person is in equilibrium, we can equate these forces:
Apparent weight at the top = Weight of the person - Centripetal force
The centripetal force can be calculated using the formula:
Centripetal force = mass x centripetal acceleration
The centripetal acceleration can be calculated using the formula:
Centripetal acceleration = (velocity^2) / radius
The velocity of the person at the top can be calculated using the circumference formula:
Velocity = (2 x π x radius) / time
Substituting the given values:
Velocity = (2 x π x 12.0 m) / 12.5 s
Once we have the centripetal acceleration, we can calculate the apparent weight at the top.
(b) At the bottom of the Ferris wheel, the person is in the lowest position. The apparent weight will still be the sum of the gravitational force and the centripetal force, but this time the gravitational force is acting upwards and the centripetal force is acting downwards.
Following the same process as above, we can calculate the apparent weight at the bottom.
To find the fractional change in apparent weight, subtract the weight at rest from the calculated apparent weight, and then divide it by the weight at rest:
Fractional change = (Apparent weight - Weight at rest) / Weight at rest
Use the above steps to calculate the fractional change at both the top and bottom of the Ferris wheel.