Understanding the Problem
* Centripetal Force: The force that causes an object to move in a circular path is called centripetal force. In this case, the fan blade is moving in a circle, and the centripetal force is what keeps it from flying off.
* Factors Affecting Force: The force exerted by the fan blade depends on the following:
* Mass (weight): The heavier the weight, the greater the force.
* Radius: The larger the radius of the circular path, the greater the force.
* Speed: The faster the rotation, the greater the force.
Calculations
1. Convert Units:
* Weight: 3 ounces = 3/16 pounds
* Radius: 15 inches = 1.25 feet
* Speed: 900 rpm = 900 revolutions per minute = 15 revolutions per second (rps)
* Angular Velocity (ω): 15 rps * 2π radians/revolution = 30π radians/second
2. Centripetal Force Formula:
* F = m * ω² * r
* F = Force (in pounds)
* m = Mass (in pounds)
* ω = Angular velocity (in radians/second)
* r = Radius (in feet)
3. Plug in Values:
* F = (3/16 pounds) * (30π radians/second)² * (1.25 feet)
* F ≈ 210.94 pounds
Important Considerations
* Simplified Model: The above calculation is a simplified model. It assumes that the fan blade is a point mass and that its motion is perfectly circular. In reality, the blade has a distributed mass and its motion might not be perfectly circular due to aerodynamic forces.
* Aerodynamic Forces: The force calculated above is the centripetal force required to keep the blade moving in a circle. However, the fan blade also experiences aerodynamic forces (lift and drag) from the air it's pushing. These forces can be significant and depend on the blade's shape, angle of attack, and airflow.
* Force Distribution: The force is not evenly distributed across the entire blade. The force will be higher at the tip of the blade and lower near the hub.
Conclusion
The simplified calculation above provides a rough estimate of the force exerted by the fan blade. In a real-world scenario, factors like blade shape, airflow, and the force distribution would need to be considered for a more accurate analysis.
