Moment of Inertia of a windmill or fan

[science4peace]

My lab partners and I did an experiment for my general physics class in which we constructed a windmill with wooden dowels. We added clay to the ends to set the concentration of mass about the ends. The distance (radius) of the clay balls from the center of the fixed axis is our control variable. We were looking for the relationship between the radius which I believe contributes to I (moment of inertia) and angular velocity. Am I right to think that a fan or windmill is just like a hoop?

Homework Equations

The Attempt at a Solution

 

[LowlyPion]

Welcome to PF.

No.

When you put the preponderance of the weight at a particular radius, jam a wad of clay at one radius, then that is acting like a hoop of that radius. But when you have an array of clay elements at different radii, then the resulting moment of inertia will need to take into account all of the contributions of all the moment elements.

 

[nlightner]

Yes, you are correct in thinking that a fan or windmill can be modeled as a hoop for the purpose of calculating the moment of inertia. In fact, the moment of inertia of a hoop is given by the equation I = MR^2, where M is the mass of the hoop and R is the radius. In your experiment, by varying the radius of the clay balls, you were effectively changing the moment of inertia of the windmill. This is because the moment of inertia is directly proportional to the square of the radius. As the radius increases, the moment of inertia also increases, resulting in a slower angular velocity for the windmill. This relationship between moment of inertia and angular velocity is known as the conservation of angular momentum, which states that the product of moment of inertia and angular velocity remains constant as long as there are no external torques acting on the system. Your experiment is a great way to demonstrate this concept in action. Keep up the good work in your physics class!