The provided image outlines a calculation of rotational inertia for both a solid disk and a ring. Here’s a step-by-step explanation of the calculations shown:

1. Rotational Inertia of the Disk ($I_\text{disk}$)

The formula for the rotational inertia of a solid disk is: $I_\text{disk} = \frac{1}{2} M R^2$ Where:

• $M = 1425 \, \text{g} = 1.425 \, \text{kg}$ (mass of the disk)
• $R = \frac{\text{diameter of the disk}}{2} = \frac{22.82 \, \text{cm}}{2} = 11.41 \, \text{cm} = 0.1141 \, \text{m}$

Calculation: $I_\text{disk} = \frac{1}{2} \times 1.425 \times (0.1141)^2$

Converting back to $\text{g} \cdot \text{cm}^2$ as seen in the image: $I_\text{disk} = \frac{1}{2} \times 1425 \times (11.41)^2$ $I_\text{disk} = 92,759 \, \text{g} \cdot \text{cm}^2$

2. Rotational Inertia of the Ring ($I_\text{ring}$)

The formula for the rotational inertia of a ring is: $I_\text{ring} = \frac{1}{2} M \left( R_\text{out}^2 + R_\text{in}^2 \right)$ Where:

• $M = 1442 \, \text{g} = 1.442 \, \text{kg}$ (mass of the ring)
• $R_\text{out} = \frac{\text{outer diameter}}{2} = \frac{12.63 \, \text{cm}}{2} = 6.315 \, \text{cm}$
• $R_\text{in} = \frac{\text{inner diameter}}{2} = \frac{10.75 \, \text{cm}}{2} = 5.375 \, \text{cm}$

Calculation: $I_\text{ring} = \frac{1}{2} \times 1442 \times \left( (6.315)^2 + (5.375)^2 \right)$ Breaking down the radii terms: $R_\text{out}^2 = (6.315)^2 = 39.87 \, \text{cm}^2, \quad R_\text{in}^2 = (5.375)^2 = 28.89 \, \text{cm}^2$ Adding these together: $R_\text{out}^2 + R_\text{in}^2 = 39.87 + 28.89 = 68.76 \, \text{cm}^2$

Substituting back: $I_\text{ring} = \frac{1}{2} \times 1442 \times 68.76$ $I_\text{ring} = 49,811 \, \text{g} \cdot \text{cm}^2$

Let me know if you'd like further clarification or help understanding the process!

Here are some related questions to explore:

1. How do uncertainties in the measurements affect the calculated values of rotational inertia?
2. What are the physical implications of the calculated values of $I_\text{disk}$ and $I_\text{ring}$?
3. Can these formulas for rotational inertia be extended to 3D objects of different shapes?
4. How does rotational inertia relate to angular acceleration and torque?
5. What real-world applications might involve using these specific calculations?

Tip: Always check units carefully when performing physics calculations, especially when converting between metric units!