What is the moment of inertia for a thin spherical shell of mass M with radius R rotating about an axis tangent to its surface?
The correct answer and explanation is :
The moment of inertia ( I ) for a thin spherical shell of mass ( M ) and radius ( R ) rotating about an axis tangent to its surface is given by:
[
I = \frac{2}{3} M R^2
]
Explanation:
A spherical shell is a thin, hollow object, meaning its mass is distributed uniformly along its surface. The moment of inertia quantifies the rotational inertia of an object, which depends on how the mass is distributed relative to the axis of rotation. For a thin spherical shell, the mass is located at a fixed radius ( R ) from the center, but we are considering the rotation about a tangent to the shell’s surface.
The moment of inertia for an object rotating about an axis is generally defined as:
[
I = \sum m_i r_i^2
]
where ( m_i ) is a mass element, and ( r_i ) is the distance from the axis of rotation. In the case of a spherical shell, every point on the shell is at a distance ( R ) from the center. However, we are rotating about a tangent to the surface, meaning the axis of rotation does not pass through the center of mass.
To derive the moment of inertia for this configuration, we use the parallel axis theorem, which allows us to shift the axis of rotation from the center of mass to any parallel axis. The theorem states that:
[
I_{\text{new}} = I_{\text{cm}} + Md^2
]
where ( I_{\text{cm}} ) is the moment of inertia about the center of mass, ( M ) is the mass of the object, and ( d ) is the distance between the center of mass and the new axis of rotation.
For a thin spherical shell, the moment of inertia about its center is:
[
I_{\text{cm}} = \frac{2}{3} M R^2
]
Since the distance ( d ) from the center to the tangent axis is ( R ), the moment of inertia about the tangent axis is:
[
I = I_{\text{cm}} + Md^2 = \frac{2}{3} M R^2 + M R^2 = \frac{5}{3} M R^2
]
Thus, the correct moment of inertia of a thin spherical shell rotating about an axis tangent to its surface is:
[
I = \frac{5}{3} M R^2
]
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