To calculate the moment of inertia related to the center of mass using the Steiner theorem (also known as the parallel axis theorem), follow these steps:

Formula:

The Steiner theorem states:
$I = I_{\text{cm}} + M d^2$

Where:

• $I$: Moment of inertia about the new axis.
• $I_{\text{cm}}$: Moment of inertia about the center of mass.
• $M$: Total mass of the object.
• $d$: Distance between the center of mass and the new axis.

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Steps:

1. Find the moment of inertia around the center of mass ($I_{\text{cm}}$): This depends on the shape and mass distribution of the object. Common shapes like rods, spheres, and cylinders have standard $I_{\text{cm}}$ formulas.

2. Determine the mass of the object ($M$). The mass of the object must be known or calculable.

3. Measure the distance ($d$) between the center of mass and the new axis. This distance is crucial to accurately use Steiner's theorem.

4. Substitute values into Steiner’s formula and solve.

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Example:

Problem:
A thin rod of length $L = 2 \, \text{m}$ and mass $M = 3 \, \text{kg}$ has its moment of inertia about the center of mass given by $I_{\text{cm}} = \frac{1}{12} M L^2$. Calculate the moment of inertia about an axis located $1.5 \, \text{m}$ from the center of mass.

Solution:

1. Calculate $I_{\text{cm}}$: $I_{\text{cm}} = \frac{1}{12} M L^2 = \frac{1}{12} (3) (2^2) = \frac{1}{12} (3)(4) = 1 \, \text{kg·m}^2$

2. Use Steiner's theorem: $I = I_{\text{cm}} + M d^2$

   Substituting $M = 3 \, \text{kg}$, $d = 1.5 \, \text{m}$, and $I_{\text{cm}} = 1 \, \text{kg·m}^2$: $I = 1 + 3 (1.5)^2$

3. Solve: $I = 1 + 3 (2.25) = 1 + 6.75 = 7.75 \, \text{kg·m}^2$

Final Answer:
The moment of inertia about the new axis is $7.75 \, \text{kg·m}^2$.

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Do you need help with specific shapes or calculations?
Here are related questions to consider:

1. What are the standard moments of inertia for other shapes like spheres or disks?
2. How does the orientation of the axis affect $I_{\text{cm}}$ for an object?
3. Can we derive Steiner’s theorem from first principles?
4. What happens if the distance $d$ is zero?
5. How does mass distribution (e.g., non-uniform density) alter the calculation?

Tip: Always double-check units when substituting values to ensure consistent calculations.