A woman rides a carnival Ferris wheel at radius 20 m, completing 4.8 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 need to determine how much time it takes for the woman to complete one full revolution on the Ferris wheel.
The number of turns the woman completes per minute is given as 4.8 turns. This means that in 1 minute, she completes 4.8 revolutions.
To find the time period per revolution, we can divide the total time taken (1 minute) by the number of revolutions.
(a) Period of motion = 1 minute / 4.8 revolutions = 0.20833 minutes/revolution
The period of motion is approximately 0.20833 minutes per revolution.
Now, let's determine the centripetal acceleration at the highest and lowest points of the Ferris wheel.
Centripetal acceleration is given by the formula:
a = (v^2) / r
where v is the linear velocity and r is the radius of the circular path.
At the highest point of the Ferris wheel, the woman's linear velocity will be at its minimum because the wheel is moving slowly.
To find the linear velocity, we can use the formula:
v = (2πr) / T
where T is the period of motion.
(b) At the highest point, the linear velocity is v = (2π * 20) / 0.20833 = 602.75 m/minute.
Now we can calculate the centripetal acceleration:
a = (602.75^2) / 20 = 18,095.31 m/minute^2
Therefore, the magnitude of the centripetal acceleration at the highest point is approximately 18,095.31 m/minute^2.
At the lowest point of the Ferris wheel, the linear velocity is at its maximum because the wheel is moving faster.
(c) At the lowest point, the linear velocity is v = (2π * 20) / 0.20833 = 602.75 m/minute.
Again, we can calculate the centripetal acceleration:
a = (602.75^2) / 20 = 18,095.31 m/minute^2
So, the magnitude of the centripetal acceleration at the lowest point is also approximately 18,095.31 m/minute^2.