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Moment of inertia of a thin spherical shell of mass M and radius R, about its diameter is
- a)MR2
- b)(1/2)MR2
- c)(2/5)MR2
- d)(2/3)MR2
Correct answer is option 'D'. Can you explain this answer?
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Moment of inertia of a thin spherical shell of mass M and radius R, ab...
Understanding Moment of Inertia
The moment of inertia (I) quantifies the rotational inertia of an object about a given axis. For a thin spherical shell, this is calculated based on its mass distribution relative to the axis of rotation.
Moment of Inertia of a Thin Spherical Shell
For a thin spherical shell of mass M and radius R, the moment of inertia about its diameter can be derived from the principles of integration or by using standard formulas for common shapes.
Key Formula
- The moment of inertia for a thin spherical shell about an axis through its center (diameter) is given by:
- I = (2/3)MR²
Explanation of the Result
- The spherical shell is uniform, meaning its mass is evenly distributed across its surface.
- When rotating about the diameter:
- Each mass element contributes to the total inertia based on its distance from the axis.
- The integration over the shell's surface leads to the factor of (2/3), which considers the geometry of a sphere.
Comparison with Other Shapes
- For comparison:
- A solid sphere has a moment of inertia of (2/5)MR².
- A solid cylinder has a moment of inertia of (1/2)MR².
- The spherical shell's unique geometry results in its specific inertia value.
Conclusion
- Thus, the moment of inertia of a thin spherical shell about its diameter is indeed (2/3)MR², confirming that the correct answer is option 'D'. This demonstrates how mass distribution and geometry significantly influence rotational characteristics.
The moment of inertia (I) quantifies the rotational inertia of an object about a given axis. For a thin spherical shell, this is calculated based on its mass distribution relative to the axis of rotation.
Moment of Inertia of a Thin Spherical Shell
For a thin spherical shell of mass M and radius R, the moment of inertia about its diameter can be derived from the principles of integration or by using standard formulas for common shapes.
Key Formula
- The moment of inertia for a thin spherical shell about an axis through its center (diameter) is given by:
- I = (2/3)MR²
Explanation of the Result
- The spherical shell is uniform, meaning its mass is evenly distributed across its surface.
- When rotating about the diameter:
- Each mass element contributes to the total inertia based on its distance from the axis.
- The integration over the shell's surface leads to the factor of (2/3), which considers the geometry of a sphere.
Comparison with Other Shapes
- For comparison:
- A solid sphere has a moment of inertia of (2/5)MR².
- A solid cylinder has a moment of inertia of (1/2)MR².
- The spherical shell's unique geometry results in its specific inertia value.
Conclusion
- Thus, the moment of inertia of a thin spherical shell about its diameter is indeed (2/3)MR², confirming that the correct answer is option 'D'. This demonstrates how mass distribution and geometry significantly influence rotational characteristics.
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Moment of inertia of a thin spherical shell of mass M and radius R, ab...
Moment of inertia:
Moment of inertia is a measure of the resistance of a body to angular acceleration about a given axis that is equal to the sum of the products of each element of mass in the body and the square of the element’s distance from the axis.
Moment of inertia of a thin spherical shell of mass M and radius R about its diameter.
I = (2/3)MR2
Additional Information
Moment of inertia of some important shapes:
Moment of inertia is a measure of the resistance of a body to angular acceleration about a given axis that is equal to the sum of the products of each element of mass in the body and the square of the element’s distance from the axis.
Moment of inertia of a thin spherical shell of mass M and radius R about its diameter.
I = (2/3)MR2
Additional Information
Moment of inertia of some important shapes:
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Moment of inertia of a thin spherical shell of mass M and radius R, about its diameter isa)MR2b)(1/2)MR2c)(2/5)MR2d)(2/3)MR2Correct answer is option 'D'. Can you explain this answer?
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