Question

The flywheel rotates with an angular velocity of ω = (0.005θ2 ) rad/s, where θ is in radians. Determine the angular accelerations when it has rotated 20 revolutions.

Answer 1

Angular acceleration of the flywheel is found by taking the derivative of the given angular velocity equation with respect to θ, multiplying by angular velocity ω, and then plugging in the value of θ after 20 revolutions, which is 40π radians.

Step-by-step explanation:

To determine the angular acceleration of the flywheel that rotates according to the equation ω = (0.005θ2) rad/s, we first need to calculate the angular position, θ, after 20 revolutions. Since 1 revolution is 2π radians, 20 revolutions is 20 × 2π = 40π radians. With θ = 40π radians plugged into the equation, the angular velocity (ω) would be (0.005)(40π)² rad/s.

However, to find the angular acceleration, we must differentiate the given angular velocity equation with respect to time (since angular acceleration is the rate of change of angular velocity with respect to time). The angular acceleration, α, can be determined using the derivative dω/dθ, multiplied by dθ/dt (which is ω itself).

We can use the chain rule for differentiation:

α = dω/dθ × dθ/dt = d(0.005θ2)/dθ × ω = 0.01θ × (0.005θ2).

At θ = 40π, the angular acceleration would then be:

α = 0.01 × 40π × (0.005 × (40π)²).

Answer 2

The angular acceleration of the flywheel when it has rotated 20 revolutions is approximately 8π rad/s.

Step-by-step explanation:

The angular velocity of a flywheel can be determined by using the equation ω = (0.005θ^2) rad/s, where θ is the angle in radians. To find the angular acceleration when the flywheel has rotated 20 revolutions, we need to convert revolutions to radians first. Since 1 revolution is equal to 2π radians, 20 revolutions is equal to 40π radians. Substituting this value into the equation ω = (0.005θ^2), we can solve for ω.

ω = (0.005(40π)^2) rad/s = (0.005(1600π)) rad/s ≈ 8π rad/s

So, the angular acceleration of the flywheel is approximately 8π rad/s.