user: Read MTH2007 Lagrangian and Hamiltonian Mechanics Cheat Sheet old bits. Can you write me a small section on moments of intertia? [Timestamp: 2024/12/04 12:26:53]
Definition: The moment of inertia (I) is a scalar value that measures how difficult it is to change the rotational motion of an object about a given axis. It depends on the mass distribution of the object relative to the axis of rotation.
1. Standard Moments of Inertia for Common Shapes
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Thin Hoop (radius ( R ), mass ( M )): [ I = MR^2 ]
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Solid Cylinder (radius ( R ), height ( h ), mass ( M )): [ I = \frac{1}{2} MR^2 ]
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Solid Sphere (radius ( R ), mass ( M )): [ I = \frac{2}{5} MR^2 ]
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Rod about Center (length ( L ), mass ( M )): [ I = \frac{1}{12} ML^2 ]
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Rod about End (length ( L ), mass ( M )): [ I = \frac{1}{3} ML^2 ]
2. Parallel Axis Theorem
The parallel axis theorem allows you to calculate the moment of inertia about any axis parallel to an axis through the center of mass. It states: [ I = I_{\text{cm}} + Md^2 ] where:
- ( I ) is the moment of inertia about the new axis,
- ( I_{\text{cm}} ) is the moment of inertia about the center of mass axis,
- ( M ) is the mass of the object,
- ( d ) is the distance between the two axes.
3. Perpendicular Axis Theorem
For planar objects, the perpendicular axis theorem relates the moment of inertia about an axis perpendicular to the plane to the moments about two orthogonal axes in the plane: [ I_z = I_x + I_y ] where:
- ( I_z ) is the moment of inertia about the perpendicular axis,
- ( I_x ) and ( I_y ) are the moments of inertia about the two axes in the plane.
4. Composite Bodies
To find the moment of inertia for composite bodies, break the object down into simpler shapes, calculate the moment of inertia for each shape, and sum them: [ I_{\text{total}} = \sum_{i} I_i ] where ( I_i ) is the moment of inertia of each component about the desired axis.
Example Calculation
For a hollow cylinder made of two thin hoops: [ I = 2 \cdot MR^2 = 2MR^2 ]
This section provides a concise overview of moments of inertia, including standard values, theorems for calculating inertia about different axes, and methods for composite bodies. [Timestamp: 2024/12/04 12:27:00]