Lesson 1: Describing Rotational Motion

Describing Rotational Motion Overview

• Focus Question: How would you describe the rotation of a hurricane?

• Conceptual Model:

  • Consider a circular object such as a DVD.

  • Mark a single point on the outer edge to track position.

  • Rotate the DVD to the left (counterclockwise).

  • When the mark completes a full loop and returns to its original starting position, the object has completed one full revolution.

Angular Displacement

• Definition: The change in the angle as an object rotates is called the object's angular displacement.

• Symbol: The Greek letter theta ($\theta$) is used to represent the angle of revolution.

• Units of Measurement for Revolutions:

  • Degree ($^\text{o}$): Defined as $\frac{1}{360}$ of a revolution. It is the standard scale marked on a conventional protractor.

  • Radian ($\text{rad}$): Defined as $\frac{1}{2\frac{\pi}{\pi}}$ or $\frac{1}{2\frac{\pi}{1}}$ ratio-based measure, specifically $\frac{1}{2\frac{\pi}{\pi}}$ of a revolution (equivalent to $\frac{1}{2\frac{\pi}{1}}$ of a full circle circumference ratio). In physics and mathematics, a radian relates the ratio of a circle's circumference to its radius.

  • In one complete revolution, a point on the edge of a wheel travels a linear distance equal to $2\pi$ times the radius of the wheel.

  • Therefore, one complete revolution equals an angle of $2\pi\,\text{rad}$, or $360^\text{o}$.

  • $1\,\text{revolution} = 2\pi\,\text{rad} = 360^\text{o}$.

  • The abbreviation for radian is $\text{rad}$.

• Sign Conventions for Rotation:

  • Counterclockwise Rotation: Designated as positive ($+\text{ve}$).

  • Clockwise Rotation: Designated as negative ($-\text{ve}$).

• Standard Fractions of a Revolution (in Radians):

  • $\theta = 0$

  • $\theta = \frac{\pi}{4}\,\text{rad}$

  • $\theta = \frac{\pi}{2}\,\text{rad}$

  • $\theta = \frac{3\pi}{4}\,\text{rad}$

  • $\theta = \pi\,\text{rad}$

  • $\theta = \frac{5\pi}{4}\,\text{rad}$

  • $\theta = \frac{3\pi}{2}\,\text{rad}$

  • $\theta = \frac{7\pi}{4}\,\text{rad}$

  • $\theta = 2\pi\,\text{rad}$

• Earth's Rotational Examples:

  • Earth turns one complete revolution ($2\pi\,\text{rad}$) in $24\,\text{h}$.

  • In $12\,\text{h}$, Earth rotates through $\pi\,\text{rad}$.

  • In $6\,\text{h}$ (one-fourth of a day), Earth rotates through an angle of $\frac{\pi}{2}\,\text{rad}$.

  • In $48\,\text{h}$, Earth rotates through an angle of $4\pi\,\text{rad}$.

  • Directional Perspective: Earth's rotation viewed from above the North Pole is counterclockwise (positive). When viewed from the South Pole, the direction appears clockwise (negative).

• Measuring Linear Distance from Angular Displacement:

  • For a rotation through an angle $\theta$, a point located at a distance $r$ from the center moves a linear arc distance $x$ given by:         $x = r\theta$

  • Variables:

    • $x$: Path distance / arc length moved ($\text{m}$).

    • $r$: Distance from the center / radius ($\text{m}$).

    • $\theta$: Angular displacement ($\text{rad}$).

  • Dimensionless Nature of Radians: Although $r$ is measured in meters ($\text{m}$), multiplying $r$ by $\theta$ (in $\text{rad}$) results in $x$ being measured in meters ($\text{m}$), not $\text{m}\cdot\text{rad}$. This is because radians represent a dimensionless ratio between arc distance $x$ and radius $r$ ($\theta = \frac{x}{r}$).

Angular Velocity

• Definition: The angular velocity of an object is defined as its angular displacement divided by the time taken to make that angular displacement.

• Symbol: Represented by the lower-case Greek letter omega ($\omega$).

• SI Unit: Radians per second ($\text{rad/s}$).

• Formula for Average Angular Velocity:     $\omega = \frac{\Delta\theta}{\Delta t}$

  • Variables:

    • $\omega$: Average angular velocity ($\text{rad/s}$).

    • $\Delta\theta$: Change in angular displacement ($\text{rad}$).

    • $\Delta t$: Time interval required ($\text{s}$).

• Instantaneous Angular Velocity:

  • Equal to the slope of a graph plotting angular position ($\theta$) versus time ($t$).

• Earth's Angular Velocity Calculation:     $\omega_E = \frac{2\pi\,\text{rad}}{(24.0\,\text{h}) \times (3600\,\text{s/h})} = 7.27 \times 10^{-5}\,\text{rad/s}$

  • Counterclockwise rotation produces a positive angular velocity.

• Relationship Between Linear Velocity and Angular Velocity:

  • If an object has an angular velocity $\omega$, the linear velocity $v$ of a point located at a distance $r$ from the axis of rotation is calculated as:         $v = r\omega$

  • Variables:

    • $v$: Linear velocity / speed ($\text{m/s}$).

    • $r$: Distance from the axis of rotation ($\text{m}$).

    • $\omega$: Angular velocity ($\text{rad/s}$).

• Linear Speed at Earth's Equator:

  • Earth's radius $r = 6.38 \times 10^{6}\,\text{m}$.

  • Angular velocity $\omega_E = 7.27 \times 10^{-5}\,\text{rad/s}$.

  • $v = (6.38 \times 10^{6}\,\text{m})(7.27 \times 10^{-5}\,\text{rad/s}) = 464\,\text{m/s}$.

• Rigid Bodies vs. Non-Rigid Bodies:

  • Rigid Body: An object whose parts are fixed relative to one another. All parts of a rotating rigid body rotate at the exact same angular velocity and rate, even though points at different radii cover different linear distances in each revolution (e.g., Earth).

  • Non-Rigid Body: An object in which different regions rotate at differing angular velocities (e.g., the Sun).

Angular Acceleration

• Definition: Angular acceleration is the change in angular velocity divided by the time required to make that change.

• Symbol: Represented by the lower-case Greek letter alpha ($\alpha$).

• SI Unit: Radians per second squared ($\text{rad/s}^2$).

• Formula for Average Angular Acceleration:     $\alpha = \frac{\Delta\omega}{\Delta t}$

  • Variables:

    • $\alpha$: Average angular acceleration ($\text{rad/s}^2$).

    • $\Delta\omega$: Change in angular velocity ($\text{rad/s}$).

    • $\Delta t$: Time interval required ($\text{s}$).

• Sign Convention: If the change in angular velocity is positive, then the angular acceleration is positive.

• Instantaneous Angular Acceleration:

  • Equal to the slope of a graph plotting angular velocity ($\omega$) as a function of time ($t$).

• Example Scenario (Car Wheel):

  • A car accelerates linearly from $0.0\,\text{m/s}$ to $25\,\text{m/s}$ in $15\,\text{s}$.

  • The car's wheels have a diameter of $0.64\,\text{m}$.

  • During this $15\,\text{s}$ interval, the angular velocity of the wheels changes from $0.0\,\text{rad/s}$ to $78\,\text{rad/s}$.

• Relationship Between Linear Acceleration and Angular Acceleration:

  • The linear acceleration $a$ of a point located at a distance $r$ from the axis of rotation is:         $a = r\alpha$

• Summary Table: Linear and Angular Quantities:

  • Displacement:

    • Linear Quantity: $x\,(\text{m})$

    • Angular Quantity: $\theta\,(\text{rad})$

    • Relationship Equation: $x = r\theta$

  • Velocity:

    • Linear Quantity: $v\,(\text{m/s})$

    • Angular Quantity: $\omega\,(\text{rad/s})$

    • Relationship Equation: $v = r\omega$

  • Acceleration:

    • Linear Quantity: $a\,(\text{m/s}^2)$

    • Angular Quantity: $\alpha\,(\text{rad/s}^2)$

    • Relationship Equation: $a = r\alpha$

Angular Frequency

• Definition: The number of complete revolutions made by an object per unit time (specifically in $1\,\text{s}$) is called angular frequency.

• Formula:     $f = \frac{\omega}{2\pi}$

• Common Units: Revolutions per minute ($\text{RPM}$).

• Real-World Application (Computer Hard Drives):

  • Standard/Inexpensive computer hard drives rotate at frequencies of $4800\,\text{RPM}$, $5400\,\text{RPM}$, and $7200\,\text{RPM}$.

  • High-performance/Advanced hard drives operate at frequencies of $10{,}000\,\text{RPM}$ or $15{,}000\,\text{RPM}$.

  • Higher rotational frequency allows faster access and storage of data.

Practice Problems

• Problem 1: What is the angular displacement of each of the following hands of a clock in $1.00\,\text{h}$? State your answer in three significant digits.

  • a. The second hand

  • b. The minute hand

  • c. The hour hand

• Problem 2: A rotating toy above a crib makes one complete counterclockwise rotation in $1\,\text{min}$.

  • a. What is its angular displacement in $3\,\text{min}$?

  • b. What is the toy's angular velocity in $\text{rad/min}$?

  • c. If the toy is turned off, does it have positive or negative angular acceleration? Explain.

Additional Practice Problems

• Problem 3: If a truck has a linear acceleration of $1.85\,\text{m/s}^2$ and the wheels have an angular acceleration of $5.23\,\text{rad/s}^2$, what is the diameter of the wheels?

• Problem 4: The truck in the previous problem is towing a trailer with wheels that have a diameter of $48\,\text{cm}$.

  • a. How does the linear acceleration of the trailer compare with that of the truck?

  • b. How do the angular accelerations of the wheels of the trailer and the wheels of the truck compare?

• Problem 5 (CHALLENGE): You replace the tires on your car with tires of larger diameter. How will the angular velocity and number of revolutions change for trips at the same speed and over the same distance?

Check Your Progress

• Problem 6 (Angular Velocity): The Moon ($r = 1.74 \times 10^{6}\,\text{m}$) rotates once on its axis in $27.3\,\text{days}$.

  • a. What is its period of rotation in seconds?

  • b. What is the frequency of its rotation in $\text{rad/s}$?

  • c. A rock sits on the surface at the Moon's equator. What is its linear speed due to the Moon's rotation?

  • d. Compare this speed with that of a person at the equator due to Earth's rotation.

• Problem 7 (Angular Displacement): A movie lasts $2\,\text{h}$. During that time, what is the angular displacement of each of the following?

  • a. The hour hand

  • b. The minute hand

  • c. The second hand

• Problem 8 (Angular Acceleration): In the spin cycle of a clothes washer, the drum turns at $635\,\text{rev/min}$. If the lid is opened, the motor is turned off. If the drum requires $8.0\,\text{s}$ to slow to a stop, what is the angular acceleration of the drum?

• Problem 9 (Angular Displacement): Do all parts of the minute hand on a watch have the same angular displacement? Do they move the same linear distance? Explain.

• Problem 10 (Critical Thinking): A CD-ROM has a spiral track that starts $2.7\,\text{cm}$ from the center of the disk and ends $5.5\,\text{cm}$ from the center. The disk drive must turn the disk so that the linear velocity of the track is a constant $1.4\,\text{m/s}$. Find the following:

  • a. The angular velocity of the disk (in $\text{rad/s}$ and $\text{rev/min}$) for the start of the track.

  • b. The disk's angular velocity at the end of the track.

  • c. The disk's angular acceleration if the disk is played for $76\,\text{min}$.