Angular Velocity Formula

Angular Velocity Formula

Angular velocity is a measure of how quickly an object rotates or revolves around a particular axis. It describes the rate of change of the angular position of the object with respect to time. In physics, angular velocity is typically represented by the Greek letter omega (ω) and is expressed in radians per second (rad/s). The formula for angular velocity is given as ω = θ/t. Learn more about angular velocity in this article by Extramarks

What is Angular Velocity?

Angular velocity is a fundamental concept in rotational motion, describing how quickly an object rotates around a specific axis. It quantifies the rate of change of the angular position of an object with respect to time. It is a vector quantity, which means it has both magnitude and direction. The direction of the angular velocity vector is determined by the right-hand rule: if you curl the fingers of your right hand in the direction of the rotation, your thumb points in the direction of the angular velocity vector.

Linear Velocity vs Angular Velocity

Sure, here’s a comparison of linear velocity and angular velocity in tabular form:

What is Angular Velocity Formula?

The angular velocity (\( \omega \)) of an object can be calculated using different formulas depending on the context. Here are the key formulas for angular velocity:

Basic Angular Velocity Formula

The angular velocity is the rate of change of the angular position (\( \theta \)) with respect to time (\( t \)):
\[ \omega = \frac{\Delta \theta}{\Delta t} \]

Where:
\( \omega \) is the angular velocity (in radians per second).
\( \Delta \theta \) is the change in angular position (in radians).
\( \Delta t \) is the change in time (in seconds).

Relation to Linear Velocity

If an object is moving in a circular path with radius \( r \) and has a linear velocity \( v \), the angular velocity can also be expressed as:
\[ \omega = \frac{v}{r} \]

Where:
\( v \) is the linear velocity (in meters per second).
\( r \) is the radius of the circular path (in meters).

For Rotational Systems

In systems where an object makes a certain number of revolutions or rotations, the angular velocity can be calculated from the frequency (\( f \)) or the period (\( T \)) of rotation:
\[ \omega = 2\pi f \]
\[ \omega = \frac{2\pi}{T} \]

Where:
\( f \) is the frequency of rotation (in hertz, or revolutions per second).
\( T \) is the period of one complete rotation (in seconds).

Angular Velocity Formula Derivation

Angular velocity (\( \omega \)) is a measure of how quickly an object rotates around a particular axis. Let’s derive the basic angular velocity formula and explore its relationship with linear velocity and other rotational quantities.

Basic Derivation

Consider an object rotating around a fixed axis. The angular position (\( \theta \)) of the object changes with time. If the object rotates from an angular position \(\theta_1\) to \(\theta_2\) in a time interval \(\Delta t\), the change in angular position is \(\Delta \theta = \theta_2 – \theta_1\).

Angular velocity (\(\omega\)) is defined as the rate of change of angular position with respect to time:

\[ \omega = \frac{\Delta \theta}{\Delta t} \]

As \(\Delta t\) approaches zero, this definition becomes:

\[ \omega = \frac{d\theta}{dt} \]

Where:
\(\omega\) is the angular velocity in radians per second (rad/s).
\(\theta\) is the angular position in radians.
\(t\) is the time in seconds.

Relationship with Linear Velocity

For an object moving along a circular path with radius \(r\), the linear velocity (\(v\)) is related to the angular velocity (\(\omega\)).

The distance (\(s\)) traveled along the circular path in time \(t\) is given by:

\[ s = r \theta \]

The linear velocity (\(v\)) is the rate of change of distance with respect to time:

\[ v = \frac{ds}{dt} \]

Since \( s = r \theta \), we can substitute and use the chain rule for differentiation:

\[ v = \frac{d(r \theta)}{dt} \]
\[ v = r \frac{d\theta}{dt} \]
\[ v = r \omega \]

Therefore, the angular velocity is:

\[ \omega = \frac{v}{r} \]

Angular Velocity in Terms of Frequency

If the object completes one full revolution, the angular displacement is \(2\pi\) radians. The frequency (\(f\)) of rotation is the number of complete revolutions per second.

The period (\(T\)) is the time taken for one complete revolution, and is related to the frequency by \( T = \frac{1}{f} \).

The angular velocity in terms of frequency and period can be derived as follows:

\[ \omega = \frac{\Delta \theta}{\Delta t} = \frac{2\pi \text{ (for one complete revolution)}}{T} \]

Since \( T = \frac{1}{f} \):

\[ \omega = 2\pi f \]

These derivations show the relationship between angular velocity, angular displacement, linear velocity, and frequency, providing a comprehensive understanding of rotational motion.

Applications of Angular Velocity

The applications of angular velocity are mentioned below:

• Rotary Machines: Angular velocity is essential in designing and analyzing rotary machines like turbines, engines, and electric motors. It helps in determining the efficiency and performance of these machines.
• Robotics: In robotics, angular velocity sensors (gyroscopes) are used for stabilizing and navigating robots, ensuring precise movements and orientation.
• Satellite Orientation: Satellites use gyroscopes to measure and control their angular velocity, ensuring correct orientation and stability in space.
• Aircraft Navigation: Angular velocity sensors are used in aircraft to maintain stability and assist in navigation, particularly during maneuvers.
• Planetary Rotation: Astronomers use angular velocity to study the rotation of planets, stars, and other celestial bodies, which is crucial for understanding their properties and behaviors.
• Spacecraft Attitude Control: Spacecraft use angular velocity measurements to maintain proper orientation and execute precise maneuvers during missions.

Solved Example on Angular Velocity Formula

Example 1: A wheel rotates 360 degrees (2π radians) in 4 seconds. What is its angular velocity?

Solution:

Given:
Angular displacement (\(\Delta \theta\)) = 2π radians
Time (\(\Delta t\)) = 4 seconds

\[ \omega = \frac{\Delta \theta}{\Delta t} \]

\[ \omega = \frac{2\pi}{4} = \frac{\pi}{2} \text{ radians per second} \]

Answer: The angular velocity is \(\frac{\pi}{2}\) radians per second.

Example 2: A car’s wheel has a radius of 0.5 meters and rotates at a speed of 10 meters per second. What is the angular velocity of the wheel?

Solution:

Linear velocity (\(v\)) = 10 meters per second
Radius (\(r\)) = 0.5 meters

\[ \omega = \frac{v}{r} \]

\[ \omega = \frac{10}{0.5} = 20 \text{ radians per second} \]

Answer: The angular velocity is 20 radians per second.

Example 3: A merry-go-round completes 5 revolutions per minute. What is its angular velocity in radians per second?

Solution:

Revolutions per minute (RPM) = 5
1 revolution = 2π radians
1 minute = 60 seconds

\[ \text{Frequency (} f \text{)} = \frac{5 \text{ rev/min}}{60 \text{ s/min}} = \frac{5}{60} \text{ rev/s} = \frac{1}{12} \text{ rev/s} \]

\[ \omega = 2\pi f \]

\[ \omega = 2\pi \times \frac{1}{12} = \frac{2\pi}{12} = \frac{\pi}{6} \text{ radians per second} \]

Answer: The angular velocity is \(\frac{\pi}{6}\) radians per second.

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