A measure of an object's resistance to changes in its rotational motion.

Mass and how that mass is distributed relative to the axis of rotation.

The perpendicular distance of each mass element dm to the axis of rotation.

The rotational inertia about an axis through the center of mass.

The perpendicular distance between the new axis and the axis through the center of mass.

Solid Sphere: Lower rotational inertia (mass closer to the center). Hollow Sphere: Higher rotational inertia (mass further from the center).

Center of Mass: Minimum rotational inertia. Parallel Axis: Higher rotational inertia (due to the parallel axis theorem).

Hoop: Higher rotational inertia (mass concentrated at the rim). Solid Disk: Lower rotational inertia (mass distributed throughout).

Point mass: $I = mr^2$. Continuous object: $I = \int r^2 dm$

Doubling Mass: Rotational inertia doubles. Doubling Distance: Rotational inertia quadruples (due to the $r^2$ term).

Sum the individual rotational inertias: $\I_{\text{tot}} = \sum I_i = \sum m_i r_i^2$

1. Imagine the solid is made of tiny masses dm. 2. Use the integral: $I = \int r^2 dm$. 3. Integrate over the entire object.

1. Identify the axis of rotation. 2. Find the rotational inertia about the center of mass ($I_{cm}$). 3. Determine the distance d between the axes. 4. Calculate $I' = I_{cm} + Md^2$.

1. Define a mass element dm. 2. Express dm in terms of spatial variables. 3. Determine the limits of integration. 4. Evaluate the integral $I = \int r^2 dm$.

1. Identify each particle's mass ($m_i$) and distance ($r_i$) from the axis of rotation. 2. Calculate each particle's rotational inertia ($I_i = m_i r_i^2$). 3. Sum the individual rotational inertias: $I_{tot} = \sum I_i$.