The moment of inertia depends on the shape of the object and how the mass is distributed relative to the axis of rotation. We can calculate the moment of inertia for each object using the following formulas:
- Hoop: The moment of inertia for a hoop rotating about its central axis is given by the formula:
- I = m * r² = 4.34 kg * (0.210 m)² = 0.191 kg·m²
- Solid Cylinder: The moment of inertia for a solid cylinder about its central axis is calculated as:
- I = (1/2) * m * r² = (1/2) * 4.34 kg * (0.210 m)² = 0.0464 kg·m²
- Solid Sphere: For a solid sphere, the moment of inertia is given by:
- I = (2/5) * m * r² = (2/5) * 4.34 kg * (0.210 m)² = 0.0364 kg·m²
- Thin Spherical Shell: The moment of inertia for a thin spherical shell is calculated as:
- I = (2/3) * m * r² = (2/3) * 4.34 kg * (0.210 m)² = 0.0596 kg·m²
To summarize, the moments of inertia for the objects are as follows:
- Hoop: 0.191 kg·m²
- Solid Cylinder: 0.0464 kg·m²
- Solid Sphere: 0.0364 kg·m²
- Thin Spherical Shell: 0.0596 kg·m²