How To Find Angular Velocity:7 Scenarios With Solved Problem

Definition and Importance of Angular Velocity

angular velocity is defined as the rate of change of angular displacement with respect to time. It measures how quickly an object rotates and is typically denoted by the Greek letter omega (ω). angular velocity is an essential parameter in rotational dynamics, as it relates to other rotational quantities like angular acceleration, angular momentum, and moment of inertia.

Image by Svjo – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 3.0.

Image by dnet – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 3.0.

how to find angular velocity — Image by Svjo – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 4.0.

angular velocity is an essential concept in physics and mechanics that describes the rate at which an object rotates around a fixed axis. Understanding angular velocity is crucial for various applications, such as analyzing the motion of rotating objects, calculating torque, or studying the behavior of pendulums. In this blog post, we will explore different methods and formulas to find angular velocity in various scenarios. So, let’s dive in!

Finding Angular Velocity from Frequency

A. Understanding the Relationship between Angular Velocity and Frequency

Frequency (f) represents the number of complete revolutions or cycles an object completes in a given time. The relationship between angular velocity (ω) and frequency is given by the formula:

$ω = 2πf$

B. Step-by-step Process to Determine Angular Velocity from Frequency

To find angular velocity from frequency, follow these steps:

Identify the frequency (f) of the rotating object.
Multiply the frequency by 2π to obtain the angular velocity (ω).

Examples:

Problem: A ceiling fan has blades that rotate at a speed of 120 revolutions per minute (rpm). Determine the angular velocity of the fan blades in radians per second (rad/s).

$\text{Step 1: Convert frequency from rpm to Hz.}$

$f (\text{in Hz}) = \frac{120 \text{ rpm}}{60 \text{ s/min}}$

$f = 2 \text{ Hz}$

$\text{Step 2: Calculate the angular velocity.}$

$\omega = 2\pi f$

$\omega = 2\pi \times 2 \text{ Hz}$

$\omega = 4\pi \text{ rad/s}$

$\text{Result:}$

$\text{The angular velocity of the ceiling fan blades is: }$

$\omega = 4\pi \text{ rad/s}$

$\text{For a numerical approximation:}$

$\omega \approx 4 \times 3.14159 \text{ rad/s}$

$\omega \approx 12.566 \text{ rad/s}$

$\text{Therefore, the fan blades have an angular velocity of approximately 12.566 radians per second.}$

Finding Angular Velocity Given Radius and Speed

A. The Role of Radius and Speed in Angular Velocity

When an object moves in a circular path, the distance traveled by a point on the object’s circumference in a given time is called linear speed (v). Angular velocity (ω) is related to linear speed and radius (r) by the formula:

$ω = \frac{v}{r}$

B. Detailed Procedure to Calculate Angular Velocity with Radius and Speed

To determine angular velocity using radius and speed, follow these steps:

Measure the radius (r) of the circular path.
Determine the linear speed (v) of the object.
Divide the linear speed by the radius to obtain the angular velocity.

Example:

Problem: A point on the edge of a merry-go-round is traveling at a linear speed (tangential speed) of 3 meters per second (m/s). If the radius of the merry-go-round is 2 meters, what is the angular velocity of the merry-go-round in radians per second (rad/s)?

$\text{Given:}$

$v = 3 \text{ m/s} \quad \text{(linear speed)}$

$r = 2 \text{ m} \quad \text{(radius)}$

$\text{The relationship between linear speed, angular velocity, and radius is:}$

$v = \omega r$

$\text{Solve for angular velocity:}$

$\omega = \frac{v}{r}$

$\omega = \frac{3 \text{ m/s}}{2 \text{ m}}$

$\omega = 1.5 \text{ rad/s}$

$\text{Therefore, the angular velocity of the merry-go-round is 1.5 radians per second.}$

Finding Angular Velocity from Linear Velocity

A. Connection between Linear Velocity and Angular Velocity

Linear velocity (v) refers to the rate at which an object moves along a straight path. The relationship between linear velocity and angular velocity is given by the formula:

$v = ωr$

B. Detailed Steps to Find Angular Velocity from Linear Velocity

To calculate angular velocity using linear velocity, follow these steps:

Measure the linear velocity (v) of the object.
Determine the radius (r) of the circular path.
Divide the linear velocity by the radius to obtain the angular velocity.

Example:

Problem: A bicycle tire has a diameter of 0.7 meters. The bicycle is moving at a linear speed of 5 meters per second along a path. What is the angular velocity of the tire in radians per second?

$\text{Given:}$

$d = 0.7 \text{ m} \quad \text{(diameter of the tire)}$

$v = 5 \text{ m/s} \quad \text{(linear speed)}$

$\text{The radius (r) is half of the diameter:}$

$r = \frac{d}{2} = \frac{0.7 \text{ m}}{2}$

$\text{The relationship between linear velocity, angular velocity, and radius is:}$

$v = \omega r$

$\text{Solve for angular velocity:}$

$\omega = \frac{v}{r} = \frac{5 \text{ m/s}}{\frac{0.7 \text{ m}}{2}}$

$\omega \approx \frac{5 \text{ m/s}}{0.35 \text{ m}}$

$\omega \approx 14.29 \text{ rad/s}$

$\text{Therefore, the angular velocity of the bicycle tire is approximately 14.29 radians per second.}$

Finding Angular Velocity from Torque and Moment of Inertia

A. Understanding Torque and Moment of Inertia in Angular Velocity

Torque (τ) is a rotational force that causes objects to rotate. The moment of inertia (I) represents an object’s resistance to changes in its rotational motion. Angular velocity can be found using torque and moment of inertia through the formula:

$τ = Iω$

B. Procedure to Determine Angular Velocity from Torque and Moment of Inertia

To find angular velocity using torque and moment of inertia, follow these steps:

Determine the torque (τ) acting on the object.
Identify the moment of inertia (I) of the object.
Divide the torque by the moment of inertia to obtain the angular velocity.

Example:

Problem: A disk with a moment of inertia of 0.2 kg·m² is initially at rest. A constant torque of 0.5 N·m is applied to the disk. After 4 seconds, what is the angular velocity of the disk?

$\text{Given:}$

$I = 0.2 \text{ kg}\cdot\text{m}^2 \quad (\text{moment of inertia})$

$\tau = 0.5 \text{ N}\cdot\text{m} \quad (\text{torque})$

$t = 4 \text{ s} \quad (\text{time})$

$\omega_0 = 0 \text{ rad/s} \quad (\text{initial angular velocity})$

$\text{Step 1: Calculate the angular acceleration:}$

$\alpha = \frac{\tau}{I}$

$\text{Step 2: Calculate the final angular velocity:}$

$\omega = \omega_0 + \alpha t$

$\text{Compute the values:}$

$\alpha = \frac{0.5}{0.2} = 2.5 \text{ rad/s}^2$

$\omega = 0 + 2.5 \times 4 = 10 \text{ rad/s}$

$\text{Therefore, the disk's angular velocity after 4 seconds is 10 rad/s.}$

Finding Angular Velocity in Specific Scenarios

A. How to Find Angular Velocity of a Rotating Object

To determine the angular velocity of a rotating object, follow the respective method discussed earlier based on the available parameters such as frequency, radius and speed, linear velocity, or torque and moment of inertia.

B. How to Find Angular Velocity in Simple Harmonic Motion

In simple harmonic motion, the angular velocity is related to the angular frequency (ω) by the equation:

$ω = \sqrt{\frac{k}{I}}$

where k represents the spring constant and I is the moment of inertia.

C. How to Find Angular Velocity of Earth

The angular velocity of the Earth can be calculated by dividing the Earth’s angular displacement by the time taken. However, it is important to note that the Earth’s angular velocity is very small, as it completes one full rotation in approximately 24 hours.

D. How to Find Angular Velocity of a Pendulum

For a simple pendulum, the angular velocity is given by the formula:

$ω = \sqrt{\frac{g}{L}}$

where g is the acceleration due to gravity and L represents the length of the pendulum.

Finding Angular Velocity without Certain Parameters

A. How to Find Angular Velocity without Time

If the time is not given, it is not possible to directly calculate angular velocity. However, if other parameters like angular displacement or frequency are provided, it may still be possible to determine the angular velocity using the relevant formulas.

B. How to Find Angular Velocity without Radius

When the radius of the circular path is unknown, it is not possible to find the angular velocity directly. However, if linear velocity or torque and moment of inertia are provided, it is still possible to calculate the angular velocity using the appropriate formulas.

Rabiya Khalid

Hi,
I am Rabiya Khalid, I have completed my masters in Mathematics. Article writing is my passion and I have been professionally writing for more than a year now. Being a science student, I have a knack for reading and writing about science and everything related to it.
In my free time, I let out my creative side on a canvas.