A woman rides a carnival Ferris wheel at radius 16 m, completing 4.2 turns about its horizontal axis every minute. What are (a) the period of the motion, and the magnitude of her centripetal acceleration at (b) the highest point and (c) the lowest point?
To find the period of motion, we can start by determining the time it takes for the woman to complete one full turn on the Ferris wheel. Since the woman completes 4.2 turns every minute, we can calculate the period using the formula:
Period = 1 / Number of turns per minute
Therefore, the period is:
Period = 1 / 4.2
To calculate the centripetal acceleration at the highest point and the lowest point, we will use the formula for centripetal acceleration:
Centripetal Acceleration = (Velocity^2) / Radius
To find the velocity, we need to determine the distance traveled in one period. Since the radius of the Ferris wheel is given as 16 m, the distance traveled in one period will be the circumference of a circle with radius 16 m:
Distance = 2 * π * Radius
The velocity is then calculated by dividing the distance traveled by the period:
Velocity = Distance / Period
Once we have the velocity, we can now calculate the centripetal acceleration at the highest and lowest points by substituting the values into the formula.
Let's calculate the values:
(a) Period:
Period = 1 / 4.2 = 0.2381 minutes
(b) Centripetal acceleration at the highest point:
To calculate velocity, we first need to find the distance traveled:
Distance = 2 * π * Radius = 2 * π * 16 m = 100.53 m
Velocity = Distance / Period = 100.53 m / 0.2381 min = 422.46 m/min
Centripetal Acceleration = (Velocity^2) / Radius = (422.46 m/min)^2 / 16 m = 11136.39 m/min^2
(c) Centripetal acceleration at the lowest point:
Using the same velocity calculated in (b):
Centripetal Acceleration = (Velocity^2) / Radius = (422.46 m/min)^2 / 16 m = 11136.39 m/min^2
Therefore, the (a) period of the motion is 0.2381 minutes, and the magnitude of the woman's centripetal acceleration is (b) 11136.39 m/min^2 at the highest point and (c) 11136.39 m/min^2 at the lowest point.