Describing Rotational Motion

Fundamentals of Angular Displacement

Angular displacement is defined as the measure of how much an object has rotated, representing a change in angle. To understand this concept, consider a circular object such as a DVD. If you mark a single point on the edge of the DVD and rotate it counterclockwise until the mark returns to its original position, the DVD has completed one full revolution. One complete revolution is equivalent to rotating through an angle of $360^\circ$. However, in the fields of mathematics and physics, the radian is the preferred unit because it relates the ratio of a circle's circumference to its radius.

In one full revolution, a point on the edge of a wheel travels a distance equal to $2\pi$ times the radius of the wheel ($C = 2\pi r$). Therefore, the radian ($\text{rad}$) is defined as $\frac{1}{2\pi}$ of a revolution. Consequently, one full revolution equals an angle of $2\pi\,\text{rad}$. The Greek letter theta ($\theta$) is the standard symbol used to represent the angle of revolution.

Units and Conventions of Rotation

Several scales are used to measure fractions of a revolution, with degrees and radians being the most prominent. A degree is defined as $\frac{1}{360}$ of a revolution and is the standard marking found on a protractor. In contrast, the radian is used extensively in scientific calculations. Transitioning between these units is essential: $1\,\text{rev} = 360^\circ = 2\pi\,\text{rad}$. The direction of rotation is also standardized: counterclockwise rotation is designated as positive ($+$), while clockwise rotation is designated as negative ($2−$).

Various fractions of a revolution can be expressed in radians relative to the zero point ($\theta = 0$). For example, half a revolution is $\theta = \pi\,\text{rad}$, one quarter is $\theta = \frac{\pi}{2}\,\text{rad}$, and three quarters is $\theta = \frac{3\pi}{2}\,\text{rad}$. More granular fractions include $\frac{\pi}{3}$, $\frac{\pi}{4}$, $\frac{\pi}{6}$, $\frac{2\pi}{3}$, $\frac{3\pi}{4}$, $\frac{5\pi}{4}$, $\frac{5\pi}{6}$, $\frac{7\pi}{4}$, and $\frac{11\pi}{6}$. Each of these angles is measured in the counterclockwise direction from the origin.

Earth's Rotational Dynamics and Directionality

Earth completes one full revolution, or $2\pi\,\text{rad}$, in a period of $24\,\text{h}$. This allows for the calculation of angular displacement over different time intervals. In $12\,\text{h}$, Earth rotates through an angle of $\pi\,\text{rad}$. In a period of $6\,\text{h}$, which is one-fourth of a day, Earth rotates through an angle of $\frac{\pi}{2}\,\text{rad}$. For a duration of $48\,\text{h}$, the total angular displacement is $4\pi\,\text{rad}$.

The sign of Earth's rotation depends on the observer's perspective. When viewed from the perspective of the North Pole, Earth's rotation is counterclockwise and therefore considered positive. Conversely, when viewed from the perspective of the South Pole, the rotation would appear clockwise, or negative.

Calculating Linear Distance from Angular Motion

To determine how far a point on a rotating object moves, one must relate the angular displacement to the distance from the center of rotation. A point on the edge of an object moves a distance of $2\pi$ times the radius in a single revolution. Generally, for a rotation through an angle ($\theta$), any point at a distance ($r$) from the center moves a linear distance ($x$) given by the formula:
$x = r\theta$

In this equation, $r$ represents the radius (measured in meters), $x$ represents the distance the point has rotated (also in meters), and $\theta$ represents the angle of rotation measured in radians. Although it might seem that multiplying meters by radians would result in a unit of $\text{m-rad}$, it does not; since radians indicate a dimensionless ratio between distance and radius, the final unit for $x$ is simply meters.

Defining and Calculating Angular Velocity

Angular velocity is the measure of how quickly an object rotates, defined as the angular displacement divided by the time required to complete that displacement. It is represented by the Greek letter omega ($\omega$). The average angular velocity of an object is given by the ratio:
$\omega = \frac{\Delta \theta}{\Delta t}$

This calculation provides the average angular velocity over a specific time interval ($\Delta t$). However, if the velocity changes over that interval, the average may not equal the instantaneous velocity. The instantaneous angular velocity is found by determining the slope of a graph plot of angular position ($\theta$) versus time ($t$). Just as counterclockwise rotation results in positive angular displacement, it also results in positive angular velocity. If an object has an angular velocity ($\omega$), the linear velocity ($v$) of a point at a distance ($r$) from the axis of rotation is calculated using:
$v = r\omega$

Earth's Angular Speed and the Rigid Body Concept

Earth's angular velocity ($\omega_E$) can be calculated by dividing the total angle of one revolution by the seconds in a day:
$\omega_E = \frac{2\pi\,\text{rad}}{(24.0\,\text{h})(3600\,\text{s/h})} = 7.27 \times 10^{-5}\,\text{rad/s}$

At Earth's equator, where the radius is approximately $6.38 \times 10^6\,\text{m/rad}$, the linear speed of a point due to rotation is:
$v = (6.38 \times 10^6\,\text{m/rad})(7.27 \times 10^{-5}\,\text{rad/s}) = 464\,\text{m/s}$

Earth is classified as a rotating rigid body. In a rigid body, all parts rotate through the same angle in the same amount of time, meaning every point rotates at the same rate, regardless of its distance from the axis. This is why points at different latitudes on Earth have different linear speeds but the same angular velocity. In contrast, the Sun is not a rigid body, as different parts of the Sun rotate at different angular velocities.

Understanding Angular Acceleration

Angular acceleration ($\alpha$) occurs when an object's angular velocity changes over time. It is defined as the change in angular velocity divided by the time interval required for that change. The formula for average angular acceleration is:
$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega_f - \omega_i}{\Delta t}$

Angular acceleration is measured in $\text{rad/s}^2$. If the change in angular velocity is positive, the angular acceleration is also positive. To find the instantaneous angular acceleration, one determines the slope of a graph of angular velocity as a function of time. The linear acceleration ($a$) of a point at a distance ($r$) from the axis is related to angular acceleration by:
$a = r\alpha$

As an example, if a car accelerates such that its wheels' angular velocity changes from $0.0\,\text{rad/s}$ to $78\,\text{rad/s}$ in $15\,\text{s}$, the wheels undergo angular acceleration. For wheels with a $0.64\,\text{m}$ diameter (radius $r = 0.32\,\text{m}$), this corresponds to the car's linear acceleration from $0.0\,\text{m/s}$ to $25\,\text{m/s}$.

Frequency and Technological Applications

Angular frequency, often synonymous with frequency ($f$), refers to the number of complete revolutions an object makes in one second. It is calculated as:
$f = \frac{\omega}{2\pi}$

A practical example of angular frequency is found in computer hard drives. The speed at which a hard drive spins determines how quickly it can access or store information. Rotation speeds for modern hard drives are often measured in revolutions per minute (RPM). Inexpensive drives typically rotate at $4800$, $5400$, or $7200\,\text{RPM}$, while more advanced, high-performance hard drives operate at speeds of $10,000$ or $15,000\,\text{RPM}$. In these contexts, angular displacement per second is effectively the same as angular velocity ($\omega$).

Summary of Linear and Angular Measures

The following table summarizes the relationships between linear and angular physical quantities:

• Displacement: Linear is $x\,\text{(m)}$; Angular is $\theta\,\text{(rad)}$; Relationship is $x = r\theta$.

• Velocity: Linear is $v\,\text{(m/s)}$; Angular is $\omega\,\text{(rad/s)}$; Relationship is $v = r\omega$.

• Acceleration: Linear is $a\,\text{(m/s}^2\text{)}$; Angular is $\alpha\,\text{(rad/s}^2\text{)}$; Relationship is $a = r\alpha$.

Questions and Discussion

How would you describe the rotation of a hurricane? A hurricane's rotation is described by its change in angle over time, specifically its angular displacement (measured in radians) and its angular velocity.

Identify the angle that Earth rotates in $48\,\text{h}$. Since Earth rotates $2\pi\,\text{rad}$ in $24\,\text{h}$, in $48\,\text{h}$ it rotates through $4\pi\,\text{rad}$.

Explain what the variables $r$, $x$, and $\theta$ represent. In the equation $x = r\theta$, $r$ represents the radius in meters ($\text{m}$), $x$ represents the linear distance of rotation in meters ($\text{m}$), and $\theta$ represents the angle of rotation in radians ($\text{rad}$).

Compare the angular velocity and angular acceleration of a rotating body. Angular velocity ($\omega$) measures how fast the angle is changing over time ($\text{rad/s}$), while angular acceleration ($\alpha$) measures how fast the angular velocity is changing over time ($\text{rad/s}^2$).