Moment of Inertia of Systems Video Lessons

Example

Problem: Part A. An object has rotational inertia I. The object, initially at rest, begins to rotate with a constant angular acceleration of magnitude ⍺. What is the magnitude of the angular momentum L of the object after time t?Express your answer in terms of I, ⍺, and t.Part B. A rigid, uniform bar with mass m and length b rotates about the axis passing through the midpoint of the bar perpendicular to the bar. The linear speed of the end points of the bar is v. What is the magnitude of the angular momentum L of the bar?Express your answer in terms of m, b, v, and appropriate constants.

FREE Expert Solution

Angular acceleration is expressed as:

$\overline{){\mathbf{\alpha }}{\mathbf{=}}\frac{{\mathbf{\omega }}_{\mathbf{f}}\mathbf{-}{\mathbf{\omega }}_{\mathbf{0}}}{\mathbf{t}}}$, where ωf is the final angular velocity, ω0 is the initial angular velocity, and t is time.

We also need to know that angular momentum is expressed as:

$\overline{){\mathbf{L}}{\mathbf{=}}{\mathbf{I}}{\mathbf{\omega }}}$, where I is the moment of inertia.

The moment of inertia of a rigid bar  is given by:

$\overline{){\mathbf{I}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{12}}{\mathbf{M}}{{\mathbf{L}}}^{{\mathbf{2}}}}$, where M is the mass of the rigid uniform bar.

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Problem Details

Part A. An object has rotational inertia I. The object, initially at rest, begins to rotate with a constant angular acceleration of magnitude ⍺. What is the magnitude of the angular momentum L of the object after time t?
Express your answer in terms of I, ⍺, and t.

Part B. A rigid, uniform bar with mass m and length b rotates about the axis passing through the midpoint of the bar perpendicular to the bar. The linear speed of the end points of the bar is v. What is the magnitude of the angular momentum L of the bar?
Express your answer in terms of m, b, v, and appropriate constants.