Suppose a wheel is initially rotating at 10.0 rad/s while undergoing constant angular acceleration reaching a speed of 30.0 rad/s after 20.0 seconds have elapsed. How long after the initial time has the wheel undergone half of the angular displacement that it will have gone through during the entire 20.0 second interval?Select one:a.10.0 sb.15.0 sc.14.2 sd.12.4 s

To solve this problem, we first need to find the angular acceleration. We can use the formula for angular acceleration, which is (final angular velocity - initial angular velocity) / time.

Given: Initial angular velocity (ωi) = 10.0 rad/s Final angular velocity (ωf) = 30.0 rad/s Time (t) = 20.0 s

Substituting these values into the formula, we get:

Angular acceleration (α) = (ωf - ωi) / t α = (30.0 rad/s - 10.0 rad/s) / 20.0 s α = 20.0 rad/s² / 20.0 s α = 1.0 rad/s²

Next, we need to find the total angular displacement for the 20.0 second interval. We can use the formula for angular displacement, which is ωit + 0.5α*t².

Substituting the given values into the formula, we get:

Total angular displacement = ωit + 0.5α*t² = 10.0 rad/s * 20.0 s + 0.5 * 1.0 rad/s² * (20.0 s)² = 200.0 rad + 0.5 * 1.0 rad/s² * 400.0 s² = 200.0 rad + 200.0 rad = 400.0 rad

Half of the total angular displacement is 400.0 rad / 2 = 200.0 rad.

Finally, we need to find the time it takes for the wheel to undergo half of the total angular displacement. We can use the formula for angular displacement and solve for time:

200.0 rad = ωit + 0.5α*t²

This is a quadratic equation in the form of at² + bt + c = 0, where a = 0.5*α, b = ωi, and c = -200.0 rad. Solving this equation for t gives two solutions, but since time cannot be negative, we discard the negative solution. The positive solution is approximately 14.2 seconds.

So, the answer is c. 14.2 s.