Moment of inertia for point particles

For particles with masses $m_j$ at perpendicular distances $r_j$ from an axis, $I=\sum_j m_jr_j^2$.

Axis dependence of moment of inertia

Moment of inertia depends on the mass distribution and the chosen rotation axis.

Moment of inertia SI units

The SI unit is kilogram-square meter, written $\text{kg}\cdot\text{m}^2$.

Kinetic energy of combined motion

For an object translating and rotating, kinetic energy can include $\frac{1}{2}Mv^2$ and $\frac{1}{2}I\omega^2$.

Idealized rise of a spinning object

If rotation is unchanged and air resistance is neglected, vertical launch speed determines maximum rise: $h=\frac{v_y^2}{2g}$.

Classical mechanicsLesson 4 of 8 in the course Fixed-Axis Rotation

Moment of Inertia, Rotational Energy, and Energy Conservation

5 flashcards · 5 quiz questions · 5 match pairs · 5 fill-in sentences · English

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Summary

This lesson develops the expression for the kinetic energy of a rigid body rotating about a fixed axis. It explains how angular velocity lets us combine the motion of all the body's parts into one useful formula.

A point on a rotating object moves along a circular path around the fixed axis. Its tangential speed depends on its distance from the axis: $v_t = r\omega$ . Points farther from the axis move faster, even though every point on a rigid body has the same angular velocity $\omega$ .

All flashcards

Moment of inertia for point particles: For particles with masses $m_j$ at perpendicular distances $r_j$ from an axis, $I=\sum_j m_jr_j^2$ .
Axis dependence of moment of inertia: Moment of inertia depends on the mass distribution and the chosen rotation axis.
Moment of inertia SI units: The SI unit is kilogram-square meter, written $\text{kg}\cdot\text{m}^2$ .
Kinetic energy of combined motion: For an object translating and rotating, kinetic energy can include $\frac{1}{2}Mv^2$ and $\frac{1}{2}I\omega^2$ .
Idealized rise of a spinning object: If rotation is unchanged and air resistance is neglected, vertical launch speed determines maximum rise: $h=\frac{v_y^2}{2g}$ .

Quiz questions

Which expression gives the moment of inertia of point particles about a specified axis?Answer: $I=\sum_j m_jr_j^2$
Why can changing the rotation axis change an object's moment of inertia?Answer: The particles' perpendicular distances from the axis can change.
When is mechanical energy conservation appropriate in a rotational-energy problem?Answer: When nonconservative energy transfers are absent or negligible.
How do you convert angular speed from revolutions per second to radians per second?Answer: Multiply by $2\pi$ .
What is the rotational kinetic energy of a rigid body rotating about a fixed axis?Answer: $K=\frac{1}{2}I\omega^2$

Based on University Physics Volume 1 by William Moebs, Samuel J. Ling, Jeff Sanny (OpenStax), licensed CC BY 4.0. Adapted into a summary, flashcards and quizzes.

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