Final answer:
The kinetic energy of the fan after 8.0 s is 9.02 J.
Step-by-step explanation:
To determine the kinetic energy of the fan after 8.0 s, we first need to find the final angular velocity of the fan. We can use the formula for torque to find the angular acceleration:
τ = I × α
Where τ is the torque, I is the moment of inertia, and α is the angular acceleration. Rearranging the formula, we have:
α = τ / I
Plugging in the given values, we have:
α = 0.11 N m / 0.034 kg m2 = 3.2353 rad/s2
Next, we can use the formula for angular velocity to find the final angular velocity:
ω = ω0 + α t
Where ω is the final angular velocity, ω0 is the initial angular velocity (which is assumed to be zero in this case), α is the angular acceleration, and t is the time. Plugging in the given values, we have:
ω = 0 + 3.2353 rad/s2 × 8.0 s = 25.8824 rad/s
Finally, we can use the formula for rotational kinetic energy to find the kinetic energy of the fan:
Krot = 0.5 I ω2
Plugging in the given values, we have:
Krot = 0.5 × 0.034 kg m2 × (25.8824 rad/s)2 = 9.02 J