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Angular and Linear Acceleration of a Wind Turbine
(a) Angular Acceleration ($ \alpha $):
We can calculate the angular acceleration using the formula:
\[ \alpha = \frac{\Delta \omega}{\Delta t} \]Given that the turbine slows down from 0.5 rev/s to 0 rev/s in 10 seconds:
\[ \Delta \omega = 0 \, \text{rev/s} - 0.5 \, \text{rev/s} = -0.5 \, \text{rev/s} \] \[ \Delta t = 10 \, \text{s} \]Now, let's calculate $ \alpha $.
(b) Centripetal Acceleration ($ a_c $):
We can calculate the centripetal acceleration using the formula:
\[ a_c = r \alpha \]Given that the length of the blades is $ r = 20 $ m, and using the calculated value of $ \alpha $, we can find $ a_c $.
(c) Total Linear Acceleration ($ a_{\text{total}} $):
At $ t = 0 $ s, the tangential acceleration is zero, so $ a_{\text{total}} $ is equal to the centripetal acceleration.