2. An electric fan having a radius of 0.5 m is running on HIGH. After the LOW button is pressed, the angular speed of the fan decreases to 83.8 rad/s in 17.5 s. the deceleration is 42.0 rad/s2. Determine
a) The initial angular speed of the fan
b) the linear speed of a point on the end of the blade when it was on HIGH
c) How many revolutions it had during the time
b) the linear speed of a point on the end of the blade when it was on HIGH
yuguyguyguy
To solve this problem, we'll use the relationship between angular speed, angular acceleration, and time. We'll also use the relationship between angular speed and linear speed. Here's how to find each of the requested quantities:
a) The initial angular speed of the fan. We're given the final angular speed, the deceleration, and the time it takes to decelerate. We can use the formula:
final angular speed = initial angular speed + (angular acceleration * time)
Rearranging the formula to solve for the initial angular speed, we have:
initial angular speed = final angular speed - (angular acceleration * time)
Substituting the given values, we get:
initial angular speed = 83.8 rad/s - (42.0 rad/s^2 * 17.5 s)
Calculating the result:
initial angular speed = 83.8 rad/s - (735 rad/s^2)
initial angular speed = -651.2 rad/s
b) The linear speed of a point on the end of the blade when it was on HIGH. The linear speed can be calculated using the formula:
linear speed = angular speed * radius
Given that the radius of the fan is 0.5 m, we can substitute the given values:
linear speed = (-651.2 rad/s) * 0.5 m
linear speed = -325.6 m/s
Note: The negative sign indicates that the linear speed is in the opposite direction of the rotation.
c) The number of revolutions the fan had during the given time. We can calculate this by finding the change in angle and dividing it by the angle corresponding to one full revolution (2π radians). The change in angle is given by:
change in angle = (final angular speed - initial angular speed) * time
Substituting the given values:
change in angle = (83.8 rad/s - (-651.2 rad/s)) * 17.5 s
change in angle = 734.4 rad/s * 17.5 s
change in angle = 12858 rad
Now, we can calculate the number of revolutions:
number of revolutions = change in angle / (2π radians)
number of revolutions = 12858 rad / (2π rad)
number of revolutions ≈ 2046.2 revolutions
So, the fan had approximately 2046.2 revolutions during the given time.