A ceiling fan has four blades, each with a mass of 0.40 kg and a length of 65 cm. Model each blade as a rod connected to the fan axle at one end. When the fan is turned on, it takes 4.45 s for the fan to reach its final angular speed of 2.3 rev/s. What torque was applied to the fan by the motor? Ignore torque due to the air.
To find the torque applied to the fan by the motor, we need to use the equation for torque:
Torque = Moment of inertia * Angular acceleration
The moment of inertia for a rod rotating around its axis is given by:
I = (1/3) * m * L²
Where:
I = moment of inertia
m = mass of the rod
L = length of the rod
In this case, we have four identical blades on the ceiling fan. Therefore, the total moment of inertia is:
I_total = 4 * (1/3) * m * L²
Now, we need to find the angular acceleration of the fan. The angular acceleration is given by:
Angular acceleration = (Final angular velocity - Initial angular velocity) / Time
In this case, the final angular velocity is 2.3 rev/s and the initial angular velocity is 0 rev/s (since the fan starts from rest). The time taken for the fan to reach its final angular velocity is 4.45 s. Thus, the angular acceleration can be calculated as:
Angular acceleration = (2.3 rev/s - 0 rev/s) / 4.45 s
Now, we have both the moment of inertia and the angular acceleration. We can substitute these values in the torque equation to find the torque applied by the motor:
Torque = I_total * Angular acceleration
Finally, substitute the values for the mass (0.40 kg), length (65 cm), final angular velocity (2.3 rev/s), and time (4.45 s) to calculate the torque applied by the motor.