The moment of inertia for a hollow sphere is a concept in rotational mechanics that describes the sphere’s resistance to changes in its angular velocity. It depends on the sphere’s mass, radius, and the thickness of its walls. The moment of inertia is crucial for understanding the dynamics of rotating systems, such as gyroscopes, flywheels, and celestial bodies like planets.
Moment of Inertia for Hollow Sphere
The moment of inertia of a hollow sphere is a measure of its resistance to angular acceleration. It depends on the sphere’s mass, radius, and the thickness of its walls.
For a hollow sphere with inner radius (r_i), outer radius (r_o), and mass (m), the moment of inertia about an axis passing through its center and perpendicular to the sphere’s surface is given by:
I = \frac{2}{5} m (r_o^2 + r_i^2)
Key Points:
- The moment of inertia increases with the mass and radii of the sphere.
- A hollow sphere has a greater moment of inertia than a solid sphere of the same mass and radius.
- The moment of inertia is independent of the thickness of the sphere’s walls.
Example:
Consider a hollow sphere with an inner radius of 5 cm, an outer radius of 10 cm, and a mass of 1 kg. Its moment of inertia about an axis passing through its center is:
I = \frac{2}{5} (1 kg) ((0.1 m)^2 + (0.05 m)^2) = 3.2 \times 10^{-3} kg m^2
Table of Moments of Inertia:
| Sphere Type | Moment of Inertia (I) |
|---|---|
| Solid sphere | (\frac{2}{5} m r^2) |
| Hollow sphere | (\frac{2}{5} m (r_o^2 + r_i^2)) |
| Thin-walled cylindrical shell | (mr^2) |
| Rectangular plate | (m(\frac{a^2 + b^2}{12})) |
| Rod about its end | ( \frac{1}{3} ml^2 ) |
Question 1:
What is the mathematical formula to calculate the moment of inertia of a hollow sphere?
Answer:
The moment of inertia of a hollow sphere is given by the formula I = (2/5) * M * (R² – r²), where I is the moment of inertia, M is the mass of the sphere, R is the outer radius, and r is the inner radius.
Question 2:
How does the thickness of the shell affect the moment of inertia of a hollow sphere?
Answer:
The moment of inertia of a hollow sphere increases with the thickness of the shell. This is because the mass of the sphere is distributed further from the center of rotation, resulting in a higher moment of inertia.
Question 3:
What is the significance of the moment of inertia in understanding the rotational motion of a hollow sphere?
Answer:
The moment of inertia provides information about the resistance of a hollow sphere to rotational acceleration. A sphere with a higher moment of inertia will require more torque to accelerate or decelerate its rotation.
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