To calculate the moment of inertia $I$ of a DVD rotating about a perpendicular axis through its center, we can treat the DVD as a disk.

The formula for the moment of inertia of a disk rotating about an axis through its center perpendicular to the plane is:

$I = \frac{1}{2} M R^2$

Where:

• $M$ is the mass of the disk (in kilograms),
• $R$ is the radius of the disk (in meters).

Step 1: Convert the given values to SI units

• Diameter of the DVD = 15 cm → Radius $R = \frac{15}{2}$ cm = 7.5 cm = 0.075 m.
• Mass of the DVD = 25 g = 0.025 kg.

Step 2: Plug the values into the formula

$I = \frac{1}{2} \times 0.025 \, \text{kg} \times (0.075 \, \text{m})^2$

$I = \frac{1}{2} \times 0.025 \times 0.005625$

$I = 0.0000703125 \, \text{kg} \cdot \text{m}^2$

So, the moment of inertia is approximately:

$I \approx 7.03 \times 10^{-5} \, \text{kg} \cdot \text{m}^2$

Summary:

The moment of inertia of the DVD about a perpendicular axis through its center is $7.03 \times 10^{-5} \, \text{kg} \cdot \text{m}^2$.

Let me know if you would like a detailed explanation or have any questions about the process!

Related Questions:

1. How does the moment of inertia change if the DVD is rotated about an axis at its edge instead of the center?
2. What is the moment of inertia for a solid sphere rotating about an axis through its center?
3. How would the moment of inertia differ for a thin ring compared to a solid disk of the same mass and radius?
4. What is the physical significance of the moment of inertia in rotational motion?
5. How do you calculate the kinetic energy of a rotating object using its moment of inertia?

Tip:

The moment of inertia depends on both the mass and the distribution of that mass relative to the axis of rotation, which is why different shapes have different formulas for calculation.