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Angle Rotated by Fan Blades
(a) Calculation using Integration:
To find the angle $ \theta $ through which the fan blades rotate in the first 8 seconds, we integrate the given angular velocity function $ \omega(t) $ over the time interval [0, 8]:
\[ \theta = \int_{0}^{8} \omega(t) \, dt \]
(b) Verification using Kinematic Equations:
We'll verify the result using the kinematic equation for angular motion:
\[ \theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2 \]
Given $ \theta_0 = 0 $ (initial angle), $ \omega_0 = \omega(0) $ (initial angular velocity), and $ \alpha = \frac{d\omega}{dt} $ (angular acceleration).
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