A ceiling fan has two different angular speed settings: 1 = 475 rev/min and 2 = 170 rev/min. What is the ratio a1/a2 of the centripetal accelerations of a given point on a fan blade?
Centripetal acceleration at a distance R and angular speed w is a = R w^2. The ratio of the centripetal accelerations, for a fixed value of R, is the square of the angular speed ratio,
a1/a2 = (w1/w2)^2 =(475/210)^2 = 5.12
To find the ratio of the centripetal accelerations of a given point on a fan blade, we can use the formula for centripetal acceleration: a = R * w^2, where R is the distance from the center of rotation and w is the angular speed.
Given that the two angular speed settings for the ceiling fan are 475 rev/min and 170 rev/min, let's denote them as w1 and w2, respectively. We want to find the ratio a1/a2 of the centripetal accelerations at a given point.
Now, since the ratio of centripetal accelerations is the square of the angular speed ratio, we can use the formula:
a1/a2 = (w1/w2)^2
Substituting the values we have:
a1/a2 = (475/170)^2
Now we can calculate the value of the ratio:
a1/a2 = (475/170)^2 = 5.12
Therefore, the ratio of the centripetal accelerations of a given point on the fan blade is 5.12.