A solid disk with a mass ok 100 kg and radius of 0.2m turns clockwise through an angular displacement 10 rad when starting from rest attain its maximum angular speed of 1 revolution every 0.5s. How much time did it take for the disk to go from rest to maximum angular speed?
To find the time it took for the disk to go from rest to maximum angular speed, we can use the formula for angular acceleration:
Angular acceleration (α) = Change in angular velocity (Δω) / Change in time (Δt)
We are given that the maximum angular speed is 1 revolution every 0.5s, which means the angular velocity (ω) is:
ω = 2π / 0.5 = 4π rad/s
Starting from rest, the initial angular velocity (ω₀) is 0 rad/s.
The change in angular velocity (Δω) is:
Δω = ω - ω₀ = 4π - 0 = 4π rad/s.
Now, we can rearrange the formula for angular acceleration to find the change in time (Δt):
Δt = Δω / α
Since we already know the change in angular displacement (Δθ) is 10 rad, we can use the formula for angular acceleration:
α = ω² / Δθ
Plugging in the values:
α = (4π)² / 10 = 16π² / 10 = 16π² / 10 rad/s²
Now we can calculate the change in time (Δt):
Δt = (4π rad/s) / (16π² / 10 rad/s²)
= (4π rad/s) * (10 rad/s² / 16π²)
= 40 / (16π) s
≈ 0.796 s
Therefore, it took approximately 0.796 seconds for the disk to go from rest to maximum angular speed.