A rotating fan completes 1200 revolutions every minute. Consider a point on the tip of a blade, at a radius of 0.11 m.
(a) Through what distance does the point move in one revolution?
m
(b) What is the speed of the point?
m/s
(c) What is the magnitude of its acceleration?
m/s2
(d) What is the period of the motion?
s
To find the answers to these questions, we need to use some basic formulas related to circular motion.
(a) The distance moved in one revolution can be found by calculating the circumference of the circle formed by the point on the tip of the blade. The formula for the circumference is C = 2πr, where r is the radius.
Substituting the given radius, we have:
C = 2π(0.11) = 0.22π ≈ 0.69 m
Therefore, the point moves approximately 0.69 meters in one revolution.
(b) To calculate the speed of the point, we divide the distance traveled in one revolution by the time taken for one revolution. The time taken for one revolution is the period of the motion, which we will determine in part (d).
Given that the fan completes 1200 revolutions per minute, we can convert this to the number of revolutions per second by dividing by 60:
1200/60 = 20 revolutions per second
Dividing the distance traveled in one revolution (0.69 m) by the number of revolutions per second (20), we get:
Speed = 0.69/20 ≈ 0.0345 m/s
Therefore, the speed of the point is approximately 0.0345 m/s.
(c) The magnitude of the acceleration can be found using the formula for centripetal acceleration, which is given by a = v^2/r, where v is the velocity and r is the radius.
Using the speed calculated in part (b) (0.0345 m/s) and the given radius (0.11 m), we have:
Acceleration = (0.0345)^2/0.11 ≈ 0.0109 m/s^2
Therefore, the magnitude of the acceleration is approximately 0.0109 m/s^2.
(d) The period of the motion refers to the time taken for one complete cycle or revolution. We can calculate it by dividing the time taken for each revolution (1/frequency) by the number of revolutions per second.
Given that the fan completes 1200 revolutions per minute (60 seconds), we find the time taken for each revolution as:
60 seconds/1200 revolutions = 1/20 ≈ 0.05 seconds
Therefore, the period of the motion is approximately 0.05 seconds.