To find the moment of inertia for different shapes, it's important to look at how mass is spread out in relation to the axis where the object will turn.

The moment of inertia, which we can call $I$, is shown with this formula:

$$I = \int r^2 \, dm$$

In this formula, $r$ is the distance from the axis of rotation to the tiny piece of mass called $dm$.

Here are some simple formulas for common shapes:

• Solid Cylinder (rotating around its center): $I = \frac{1}{2} m r^2$

• Solid Sphere (rotating around its center): $I = \frac{2}{5} m r^2$

• Hollow Sphere (rotating around its center): $I = \frac{2}{3} m r^2$

• Rectangular Plate (rotating around one edge): $I = \frac{1}{3} m L^2$

In these formulas:

• $m$ is the total mass of the shape.
• $r$ or $L$ are measurements related to the size of the shape.

How to Calculate Moment of Inertia:

1. Identify the Shape: Figure out which shape you are working with.

2. Determine Axis of Rotation: Decide which line the shape will spin around. This is key for your calculations.

3. Use the Formula: Substitute the values for mass and size into the right formula.

4. Evaluate: If you are using the integral formula, set it up properly and calculate the result for the whole shape.

Using these steps makes calculating the moment of inertia straightforward and accurate for different shapes.