Consider a person in an amusement park riding a carousel with a radius of 10 m The ride completes 5 revolutions about its positive x-axis (viewed from the top) every minute. Assume that the person is seating at the edge of the carousel.
a. What is the period (the time it takes to complete 1 revolution) of the motion?
b. Find the magnitude of the centripetal acceleration of the person.
c. Determine the direction of the acceleration at the positive vertical axis.
d. How about the magnitude of the acceleration at the positive vertical axis.
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Answer:
a. The period (T) of the motion is the time it takes to complete one revolution, which can be calculated using the frequency (f) of the motion:
f = 5 revolutions / minute
T = 1 / f = 1 / (5/minute) = 0.2 minutes/revolution
Therefore, the period of the motion is 0.2 minutes/revolution.
b. The centripetal acceleration (ac) of the person is given by:
ac = v^2 / r
where v is the linear speed of the person and r is the radius of the carousel. The linear speed of the person can be found from the distance traveled per revolution (circumference of the circle with radius r) and the period T:
v = 2πr / T = 2π(10 m) / 0.2 min = 62.83 m/min
Converting this to meters per second:
v = 62.83 m/min * (1 min / 60 s) = 1.05 m/s
Substituting the values:
ac = v^2 / r = (1.05 m/s)^2 / (10 m) = 0.11025 m/s^2
Therefore, the magnitude of the centripetal acceleration of the person is 0.11025 m/s^2.
c. The direction of the acceleration at the positive vertical axis is downward, towards the center of the circle. This is because the centripetal acceleration is always directed towards the center of the circle.
d. The magnitude of the acceleration at the positive vertical axis can be found by decomposing the centripetal acceleration vector into its vertical and horizontal components. The horizontal component does not contribute to the acceleration at the positive vertical axis, so we can focus on the vertical component:
a_vertical = ac * cosθ
where θ is the angle between the acceleration vector and the positive vertical axis. At the highest point of the motion (when the person is at the top of the circle), θ is equal to 90 degrees, so:
a_vertical = ac * cos(90°) = 0
Therefore, the magnitude of the acceleration at the positive vertical axis is zero at the highest point of the motion.