When a ceiling fan rotating with a angular speed of 2.60 rad/s is turned off, a frictional torque of 0.221 N*m slows it to a stop in 5.80 s. What is the moment of inertia of the fan?
Question
When a ceiling fan rotating with a angular speed of 2.60 rad/s is turned off, a frictional torque of 0.221 N*m slows it to a stop in 5.80 s. What is the moment of inertia of the fan?
Solution
To solve this problem, we need to use the formula for torque (τ) which is related to moment of inertia (I) and angular acceleration (α) as follows:
τ = I * α
We know the torque (τ) is 0.221 N*m. We also know the fan comes to a stop, so the final angular velocity (ωf) is 0 rad/s. The initial angular velocity (ωi) is 2.60 rad/s and the time (t) it takes to come to a stop is 5.80 s.
We can find the angular acceleration (α) using the formula:
α = (ωf - ωi) / t
Substituting the given values:
α = (0 rad/s - 2.60 rad/s) / 5.80 s = -0.448 rad/s²
The negative sign indicates that this is a deceleration, as expected.
Now we can find the moment of inertia (I) by rearranging the torque formula:
I = τ / α
Substituting the given values:
I = 0.221 Nm / -0.448 rad/s² = -0.493 kgm²
The negative sign indicates that the direction of the moment of inertia is opposite to the direction of the angular acceleration, which makes sense in this context. So, the moment of inertia of the fan is 0.493 kg*m².
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