7.1 Describing Circular and Rotational Motion

Rotational Motion Overview

• Spinning Objects: Examples like wind turbine blades; outer parts move faster than inner.

• Rotational Motion Definition: Motion of objects around an axis; requires new concepts.

Angular Position

• Definition: Angular position ($\theta$) measures the position of a particle in circular motion, counterclockwise from the positive x-axis.

• Units: Measured in radians (rad) rather than degrees; $\theta = \frac{s}{r}$ where $s$ is arc length and $r$ is radius.

• Conversion: 1 rev = 360 degrees = $2\pi$ rad.

Angular Displacement and Velocity

• Angular Displacement ($\Delta\theta$): Change in angular position over a time interval.

• Angular Velocity ($\omega$): Defined as $\omega = \frac{\Delta\theta}{\Delta t}$; unit is rad/s.

• Uniform Circular Motion: Constant $\omega$, angular displacement changes by the same amount each second.

Relationships with Linear Motion

• Angular Displacement Equation: $\Delta \theta = \omega \Delta t$ for uniform circular motion.

• Speed Relationship: Linear speed ($v$) relates to angular speed ($\omega$) by $v = \omega r$.

Additional Concepts

• Graphs: Angular position vs time graphs; angular velocity is the slope of these graphs.

• Angular Speed and Period: $\omega = \frac{2\pi}{T}$ where $T$ is period; also relates to frequency ($f$): $\omega = 2\pi f$.

Examples

• Angular Velocity Calculation: Example with a ball rolling inside a wheel; demonstrated calculations based on revolutions per minute.

• Wind Turbine Speed: Determined speed at various points based on angular speed; illustrated real-world application of concepts.

Conclusion

• Cultural Note: Clocks move clockwise due to historical conventions stemming from sundials affected by Earth's rotation.