4.5 Angular Displacement

Introduction

In the study of motion, while linear displacement describes an object's change in position along a straight line, angular displacement describes its change in orientation as it rotates around a central point or axis. It is the rotational analog of linear displacement and is a fundamental quantity for analyzing the kinematics of spinning or orbiting objects.

Scalar and Vector Quantities→

Definition

Angular displacement ($\theta$) is defined as the angle through which a point or line has been rotated in a specified direction about a specified axis. It measures the change in the angular position of an object.

• For a point on a spinning object: It is the angle swept out by the line connecting the point to the center of rotation.

Formula Relating Arc Length and Angular Displacement

Angular displacement provides the link between the linear distance traveled along a circular path (the arc length) and the radius of that path.

The formula is: $\theta =\frac{S}{r}$

Where:

• $\theta$ is the angular displacement in radians.
• $S$ is the arc length — the linear distance traveled along the circular path.
• $r$ is the radius of the circular path.

Rearranging: $S=r\theta$

This formula shows that for a given radius, angular displacement is directly proportional to arc length.

Worked Example: A particle moves along a circular arc of radius $0.5\text{ m}$ and covers an arc length of $1.5\text{ m}$. Find its angular displacement. $\theta =\frac{S}{r}=\frac{1.5}{0.5}=3\text{ rad}$

Units of Angular Displacement

Angular displacement can be measured in several units, but the standard SI unit is the radian.

• Radians (rad): The SI unit. One radian is the angle subtended at the center of a circle by an arc whose length equals the radius ($S=r$). Since the circumference of a circle is $2\pi r$, a full circle contains $2\pi$ radians.
• Degrees (${}^{\circ }$): A full circle is divided into $360^{\circ }$.
• Revolutions (rev): One full revolution equals one complete circle.

Supplementary Units→

Unit Conversions

The relationships between these units are crucial for calculations: $1\text{ revolution}=360^{\circ }=2\pi \text{ radians}$

Key conversion factors:

• Degrees → Radians: ${\theta }_{\text{rad}}={\theta }_{\text{deg}}\times \frac{\pi }{180^{\circ }}$
• Radians → Degrees: ${\theta }_{\text{deg}}={\theta }_{\text{rad}}\times \frac{180^{\circ }}{\pi }$

A useful value: $1\text{ rad}=\frac{180^{\circ }}{\pi }\approx 57.3^{\circ }$

Worked Example: Convert $270^{\circ }$ to radians. $\theta =270\times \frac{\pi }{180}=\frac{3\pi }{2}\text{ rad}$

Dimensions of Angular Displacement

Since $\theta =\frac{S}{r}$ is a ratio of two lengths: $[\theta ]=\frac{[L]}{[L]}=[M^0{L}^0{T}^0]$

Angular displacement is therefore a dimensionless quantity.

Vector Nature

Angular displacement is a vector quantity under certain conditions.

• Magnitude: The size of the angle of rotation.
• Direction: Determined by the right-hand rule — curl the fingers of your right hand in the direction of rotation; your thumb points in the direction of the angular displacement vector.
• By convention, counter-clockwise rotation is positive and clockwise is negative.

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