A fan rotates from rest and accelerates 1 rad per square second for 1 second
find the angular velocity
if 1 round= 2 pi find the complete no of rounds the fan rotates at constant velocity during second half of its motion
To find the angular velocity of the fan, we need to use the formula for angular acceleration:
Angular acceleration (α) = Change in angular velocity (ω) / Time taken (t)
Given that the fan starts from rest (initial angular velocity, ω0 = 0) and accelerates at a rate of 1 rad/s^2 for 1 second (t = 1 s), we can substitute these values into the formula:
1 rad/s^2 = ω / 1 s
Now, solving for ω:
ω = 1 rad/s^2 * 1 s = 1 rad/s.
Therefore, the angular velocity of the fan is 1 rad/s.
Now, let's calculate the number of complete rounds the fan rotates at a constant velocity during the second half of its motion.
We know that the fan starts from rest and accelerates at a rate of 1 rad/s^2 for 1 second. So, during the first half-second, it covers half of its total angular displacement. In this half-second, the fan will cover:
θ = (1/2) * α * t^2
= (1/2) * 1 rad/s^2 * (0.5 s)^2
= 0.125 rad.
During the second half of the motion, the fan moves at a constant velocity. Since the fan has already rotated 0.125 rad in the first half-second, it needs to rotate an additional 0.125 rad during the second half-second to make a complete revolution.
Therefore, the total number of complete rounds the fan rotates at constant velocity during the second half of its motion is:
Number of complete rounds = (Total angular displacement) / (2π)
= (0.125 rad + 0.125 rad) / (2π)
= 0.25 rad / (2π)
≈ 0.0398 rounds.
Hence, the fan rotates approximately 0.0398 complete rounds at constant velocity during the second half of its motion.