PHYS 218 Chapter 9

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Rotation of Rigid bodies

Examples:

• CDs
• Fans
• Circular Saws
• Wheels
• Gyroscopes

Rotating objects also have energy. We can express this as a function of the angle (in radians) it makes with a coordinate axis centered at the axis of rotation.

Angular Velocity

Change in angle over time. Represented as Greek letter omega (ω)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega = \frac{\theta_f - \theta_i}{t_f-t_i} = \frac{d\theta}{dt}}

Units in radians/second or just /second, since radians have no units.

ω is actually a vector:

• magnitude: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left| \frac{d\theta}{dt} \right|}
• direction: right-hand rule (perpendicular to plane of rotation)

Angular Acceleration

Change in angular velocity over time. Represented as Greek letter alpha (α)

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = \frac{d\mathbf{\omega}}{dt}=\frac{d^2\theta}{dt^2}}

Angular acceleration also follows the right-hand rule (perpendicular to plane of rotation).

Dynamics with Angular Variables

${\begin{aligned}\omega (t)&=\omega _{0}+\alpha t\\\theta (t)&=\theta _{0}+\omega _{0}t+{\frac {1}{2}}\alpha t^{2}\\\omega ^{2}(t)={\omega _{0}}^{2}2\alpha (\theta -\theta _{0})\end{aligned}}$

Point Movement Inside a Body

Velocity

$\left|\mathbf {v} _{p}\right|=\left|\mathbf {r} _{p}\right|\cdot \mathbf {\omega }$

Using the Cross Product:

$\mathbf {v} _{p}=\mathbf {r} _{p}\times \mathbf {\omega }$

Acceleration

${\begin{aligned}a_{\mathrm {tangent} }&=r_{p}\alpha \\a_{\mathrm {radial} }&={\frac {\mathbf {v} ^{2}}{r}}=r_{p}\omega \end{aligned}}$

Energy in Rotational Motion

Imagine a rigid system of particles that rotates about the origin. What is the total kinetic energy?

where $r_{i}$ is the distance from the axis to the particle.

For one particle:

$K_{i}={\frac {1}{2}}m_{i}{v_{i}}^{2}={\frac {1}{2}}m_{i}(r_{i}\omega )^{2}={\frac {1}{2}}I\omega ^{2}$

Sum over all particles for total energy:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \sum K_i = \sum \frac{1}{2}m_i(r_i\omega)^2 = \frac{\omega^2}{2} \underbrace{\sum m_i{r_i}^2}_\mathrm{moment\ of\ inertia} = \frac{\omega^2}{2} I_a}

Moment of Inertia (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} ) is defined below. In last form, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} represents that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} is taken with respect to the axis of rotation (Moment of inertia can be different for different axes)

Moment of Inertia

Represented as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} .

For a discrete system of particles (separate and countable):

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I=\sum m_i\left|\mathbf{r}_i\right|^2}

For a continuum density (ρ) of particles:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I=\int_V \rho(r) r^2\ dV}

Moment of inertia depends on the mass distribution and the position of the origin (centered at rotational axis).

When two objects stick together, their total moment of inertia is the sum of their individual moments of inertia

Moments of Inertia for Common Objects

(See Table 9.2 on P. 299)

Example

A disk and a ring have the same mass. In the ring, all of the mass is concentrated farther away from the axis of rotation. Therefore, the ring has a higher moment of inertia than the disk and will be harder to rotate.

Summary

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} v_\perp & = \omega r \\ a_\perp & = \alpha r \\ K & = \frac{1}{2} \omega^2 I \\ I & = \sum m_i{r_i}^2 \end{align}}

Parallel Axis Theorem

Given the moment of inertia for an axis that passes through the center of mass (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_{cm}} ) of an object of mass Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} , the moment of inertia for a parallel axis (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_A} ) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} meters away is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_A = I_{cm} + md^2\,\!}

Corollary

If the distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} is significantly large, the object behaves like a particle rather than a solid body.

Example

Suppose we have a disk with a moment of inertia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_A} and a nut with moment of inertia Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I_N} :

If we glue the nut onto the disk at a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = I_A + I_{N/A} = \frac{1}{2} m_dr_d^2 + \frac{1}{2}m_n(r_1+r_2)^2 + m_nd^2\,\!}