Angular Displacement (College Board AP® Physics 1: Algebra-Based)

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Katie M

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Physics

Angular displacement

Angular position

Angular position is defined as:

The rotational location (angle) of a point on a rigid system, relative to a reference point

In linear motion, the position of an object is measured relative to the origin of a coordinate system
In rotational motion, the angular position of a point on a rigid system is measured relative to the axis of rotation

Diagram of a circle representing a rotating system with an axis at O. Labels indicate position vector, radius, point on the rotating system, angular position, arc length, and direction of rotation. — The angular position of a point on a rigid system is measured relative to the axis of rotation, which is the horizontal x-axis. As the point moves along its circular path, the position vector r sweeps out an angle θ, which increases in the counterclockwise direction.

The angular position of a point on a rigid system $θ$ is measured relative to a reference line
As the point moves, the only quantity that changes is $θ$
The reference line could be a coordinate plane or another position vector
- For example, point A is at an angle of 30° with respect to the x-axis
- Or, point P on an extended rigid object rotates clockwise by 75° with respect to its initial position vector

Rotation of an extended object about a fixed reference axis

Diagram showing a circular motion with points P at initial and final positions, connected to a fixed reference axis O, illustrating angular position θ. — When a rigid, extended object rotates about an axis through the point O, its angular position is described by the angle through which a point P has rotated about that axis relative to a fixed reference axis

The relative nature of a point's angular position gives it a directional component
As a result, angular position is a vector quantity with both magnitude and direction

Angular displacement

Angular displacement is defined as:

The change in angle through which a point on a rigid system rotates about a specified axis

The angular displacement of a rotating rigid system describes the difference in angular position between a point's starting position vector and its finishing position vector
This can be described by the equation:

$∆ θ = θ - θ_{0}$

Where:
- $∆ θ$ = angular displacement, in $rad$
- $θ$ = final angular position, in $rad$
- $θ_{0}$ = initial angular position, in $rad$

Defining angular displacement

Diagram of a circle with center O. Points A and B are on the circle's circumference. A red angle Δθ, green angles θ0, and θ are shown. The x-axis is labeled. — **When a point on a rotating rigid body moves from initial position A to final position B, the angular displacement Δθ is equal to the difference between the initial and final angular positions**

Direction of angular displacement

Like linear displacement, angular displacement is a vector quantity with both magnitude and direction
- The direction of a system's angular displacement can be described as clockwise or counterclockwise with respect to a given axis of rotation
- Mathematically, either direction can be assigned as positive only if the opposing direction is negative
For example, consider a disc rotating about an axis of rotation at its center, where counterclockwise is defined as the positive direction
- If the disc rotates counterclockwise, its angular displacement is positive
- If the disc rotates clockwise, its angular displacement is negative

Three diagrams showing rotation of a disc with a reference line. Left: point P; center: counterclockwise path is positive θ; right: clockwise path is negative θ. — If counterclockwise is defined as the positive direction, then when the disc rotates counterclockwise, its angular displacement relative to the reference line is positive, and negative when it rotates clockwise

Units of angular displacement

Angular displacement is measured in radians
A radian, or rad, is defined as:

The angle subtended at the center of a circle by an arc equal in length to the radius of the circle

It can be expressed as:

$θ = \frac{s}{r}$

Where:
- $θ$ = angle subtended from the center of the circle, in $rad$
- $s$ = arc length, in $m$
- $r$ = radius of the circle, in $m$

Definition of a radian

Illustration showing a circle with radius r, an angle Δθ, and an arc length S. It states "1 rad when S = r" indicating radians measurement. — **When the angle is equal to one radian, the length of the arc (S) is equal to the radius (r) of the circle**

Any angle in degrees can be converted into radians using the equation:

$θ (rad) = θ (°) \times \frac{π}{180 °}$

Radians are commonly written in terms of π, where:
- 360° is equivalent to $2 π$
- 270° is equivalent to $\frac{3 π}{2}$
- 180° is equivalent to $π$
- 90° is equivalent to $\frac{π}{2}$

Examiner Tip

You should be aware that a radian is not technically a unit. It is a dimensionless ratio, as the units of arc length and radius cancel. This means that an angle described in radians has no unit and does not need to be converted from one unit to another. However, we often write “rad” after an angle measured in radians to remind ourselves that the quantity describes an angle.

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Angular Displacement (College Board AP® Physics 1: Algebra-Based)

Study Guide

Angular displacement

Angular position

Rotation of an extended object about a fixed reference axis

Angular displacement

Defining angular displacement

Direction of angular displacement

Units of angular displacement

Definition of a radian

Examiner Tip

You've read 0 of your 10 free study guides

Unlock more, it's free!

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Author: Katie M