How to Calculate Rotational Inertia for Different Objects?

[ahuebel]

I would like to further understand rotational inertia. I understand that for a point mass, I = MR^2 and for a continuous object it is basically the sum of all "little" MR^2 for each element of that object. I get a little fuzzy when actually solving for I for an object. For example, if we have a cone, the mass M of the cone is the density * volume or rho*(1/3)pi*r^2. So to find inertia we take the integral of the product of mass and R^2 but over what interval? My book says to take it over the volume but I am not 100% sure what that means. Would it be the triple integral dx dy dz (or more easily dr d(theta) dz)? If so, what would be the integrand?

TIA for any help

 

[arildno]

First, always be clear about where the axis of rotation is, relative to the position of the object.
If we let the z-axis denote the rotation axis, then the moment of inertia of the object is given by $$I=\int_{V}\rho(x^{2}+y^{2})dV$$
where x,y are orthogonal coordinates in a plane defined by the rotation axis.

 

[charng]

Sure, I would be happy to provide some information on rotational inertia and how to calculate it for different objects. Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to rotational motion. It is similar to mass in linear motion, but instead of measuring an object's resistance to linear motion, it measures its resistance to rotational motion.

When solving for rotational inertia, the first step is to determine the axis of rotation. This is the imaginary line around which the object will rotate. For a point mass, the rotational inertia is simply the product of the mass (M) and the square of the distance (R) from the axis of rotation. However, for a continuous object, it is necessary to consider the distribution of mass throughout the object.

As you mentioned, for a continuous object, the rotational inertia is the sum of all the "little" MR^2 for each element of the object. This can be expressed mathematically as an integral, which represents the summation of infinitely small elements. In the case of a cone, the integral would be taken over the volume of the cone. This means that the limits of integration would be from the base of the cone (where r=0) to the apex (where r=R) and the integrand would be the product of the density (ρ) and the square of the distance (r) from the axis of rotation.

To solve this integral, you can use either a triple integral in Cartesian coordinates (dx dy dz) or a cylindrical integral (dr dθ dz). The choice of integration method will depend on the shape of the object and the axis of rotation. In the case of a cone, using cylindrical coordinates would be more convenient.

I hope this helps to clarify the process of calculating rotational inertia for an object. If you have any further questions, please don't hesitate to ask.