wf^2=wi^2+2ad
d= 10*2PI
d= 10*2PI
ω = ω₀ + αt
Where:
ω is the final angular velocity
ω₀ is the initial angular velocity (in this case, 0 because the wheel starts from rest)
α is the angular acceleration (given as 4.0 rad/s^2)
t is the time taken to reach the final angular velocity (which we need to find)
Since the wheel starts from rest, its initial angular velocity ω₀ is 0. We need to find the time t it takes for the wheel to make 10 revolutions.
Since 1 revolution is equal to 2π radians, 10 revolutions is equal to 20π radians. Therefore, our final angular displacement θ is 20π radians.
To find the time t, we can use the formula for angular displacement:
θ = ω₀t + 0.5αt²
Since ω₀ is 0 in this case, the equation simplifies to:
θ = 0.5αt²
Plugging in the values, we have:
20π = 0.5 * 4.0 * t²
Now, we can solve for t:
t² = (20π) / (0.5 * 4.0)
t² = 10π
Taking the square root of both sides, we get:
t ≈ √(10π)
Using a calculator, we find that t ≈ 5.02 seconds.
Now, we can substitute this value of t into the original equation to find the final angular velocity ω:
ω = ω₀ + αt
ω = 0 + 4.0 * 5.02
Simplifying, we get:
ω ≈ 20.08 rad/s
Therefore, when the wheel has made 10 revolutions, its angular velocity is approximately 20.08 rad/s.