a disk of radius 10 cm speeds up from rest. it turns 60 radians reaching an angular velocity of 15 rad/s. what was the angular acceleration?
b. how long did it take the disk to reach this velocity?
To find the angular acceleration, we can use the following formula:
Angular acceleration (α) = (final angular velocity (ω) - initial angular velocity (ω₀)) / time (t)
Given:
Radius (r) = 10 cm
Angle (θ) = 60 radians
Final angular velocity (ω) = 15 rad/s
Initial angular velocity (ω₀) = 0 rad/s
a) Finding the angular acceleration (α):
To find the angular acceleration, we can rearrange the formula:
α = (ω - ω₀) / t
We are given ω = 15 rad/s and ω₀ = 0 rad/s. We need to find the time (t).
b) Finding the time (t) taken to reach the final velocity:
Using the formula:
θ = ω₀ * t + (1/2) * α * t²
We are given θ = 60 radians, ω₀ = 0 rad/s, α (which we need to find), and we need to find t.
Let's solve for α first:
α = (ω - ω₀) / t = (15 rad/s - 0 rad/s) / t = 15 rad/s / t
Now, let's solve for t using the second equation:
θ = ω₀ * t + (1/2) * α * t²
Plugging in the known values:
60 radians = 0 rad/s * t + (1/2) * (15 rad/s / t) * t²
Simplifying:
60 = (15/2) * t
Dividing both sides by (15/2):
t = 8 seconds
a) The angular acceleration (α) is:
α = (ω - ω₀) / t = (15 rad/s - 0 rad/s) / 8 s ≈ 1.875 rad/s²
b) It took the disk 8 seconds to reach the final velocity.