How To Find Angular Acceleration From Angular Velocity: Problem And Examples

In the field of physics and mechanics, understanding rotational motion is essential. One important concept in rotational motion is angular acceleration, which relates to the rate of change of angular velocity. In this blog post, we will explore how to find angular acceleration from angular velocity, step by step. We will also cover special cases and related concepts, providing clear explanations and examples along the way.

How to Calculate Angular Acceleration from Angular Velocity

angular acceleration from angular velocity 1

The Mathematical Formula

angular acceleration from angular velocity 3

The formula to calculate angular acceleration $(alpha$ ) from angular velocity $(omega$ ) is given by:

$alpha = frac{{Delta omega}}{{Delta t}}$

Here, $Delta omega$ represents the change in angular velocity, and $Delta t$ represents the change in time.

Step-by-Step Process

To find the angular acceleration from angular velocity, follow these steps:

Identify the initial angular velocity $(omega_i$ ) and the final angular velocity $(omega_f$ ).
Determine the time interval $(Delta t$ ) during which the change in angular velocity occurs.
Calculate the change in angular velocity $(Delta omega$ ) by subtracting the initial angular velocity from the final angular velocity: $Delta omega = omega_f - omega_i$ .
Divide the change in angular velocity by the change in time to obtain the angular acceleration: $alpha = frac{{Delta omega}}{{Delta t}}$ .

Worked out Example

Image by User:Cdang – Wikimedia Commons, Wikimedia Commons, Licensed under CC BY-SA 4.0.

Let’s work through an example to better understand how to find angular acceleration from angular velocity.

Example: A wheel starts with an initial angular velocity of 2 rad/s and accelerates uniformly to a final angular velocity of 8 rad/s over a time interval of 4 seconds. Calculate the angular acceleration.

Solution:
Given:
Initial angular velocity $(omega_i$ ) = 2 rad/s
Final angular velocity $(omega_f$ ) = 8 rad/s
Time interval $(Delta t$ ) = 4 s

To find the angular acceleration $(alpha$ ), we can use the formula: $alpha = frac{{Delta omega}}{{Delta t}}$ .

First, calculate the change in angular velocity $(Delta omega$ ):
$Delta omega = omega_f - omega_i = 8 , text{rad/s} - 2 , text{rad/s} = 6 , text{rad/s}$ .

Next, divide the change in angular velocity by the change in time:
$alpha = frac{{Delta omega}}{{Delta t}} = frac{{6 , text{rad/s}}}{{4 , text{s}}} = 1.5 , text{rad/s}^2$ .

Therefore, the angular acceleration of the wheel is $1.5 , text{rad/s}^2$ .

Special Cases in Finding Angular Acceleration

How to Find Angular Acceleration without Time

In some cases, the time interval $(Delta t$ ) may not be given. However, it is still possible to find the angular acceleration using other known quantities.

If the initial angular velocity $(omega_i$ ), final angular velocity $(omega_f$ ), and the change in angular displacement $(Delta theta$ ) are known, the angular acceleration can be found using the following formula:

$alpha = frac{{(omega_f)^2 - (omega_i)^2}}{{2 cdot Delta theta}}$

Where $Delta theta$ represents the change in angular displacement.

How to Find Angular Acceleration from Angular Velocity and Radius

In situations where the angular velocity $(omega$ ) and radius $(r$ ) are known instead of time, the angular acceleration can be determined using the following formula:

$alpha = frac{{omega^2}}{{r}}$

Where $r$ represents the radius.

How to Find Angular Acceleration from Angular Velocity and Time

If the angular velocity $(omega$ ) and tangential acceleration $(a_t$ ) are given, the angular acceleration can be calculated using the formula:

$alpha = frac{{a_t}}{{r}}$

Where $r$ is the radius.

Related Concepts

How to Find Tangential Acceleration from Angular Velocity

To find the tangential acceleration $(a_t$ ) from angular velocity $(omega$ ) and radius $(r$ ), you can use the formula:

$a_t = omega cdot r$

This formula relates the linear velocity $(v$ ) to the angular velocity $(omega$ ) and radius $(r$ ), as tangential acceleration is the rate of change of linear velocity.

How to Find Linear Acceleration from Angular Velocity

Linear acceleration $(a$ ) can be determined from angular velocity $(omega$ ) and radius $(r$ ) using the formula:

$a = alpha cdot r$

Where $alpha$ represents the angular acceleration.

How to Calculate Centripetal Acceleration from Angular Velocity

Centripetal acceleration $(a_c$ ) can be calculated using the formula:

$a_c = frac{{v^2}}{{r}} = omega^2 cdot r$

Here, $v$ represents the linear velocity and $r$ is the radius.

Understanding how to find angular acceleration from angular velocity is crucial for analyzing rotational motion. By following the steps outlined in this blog post, you can calculate angular acceleration accurately. Remember to consider special cases and related concepts to gain a comprehensive understanding of this topic.

How can you find the constant angular acceleration from the given angular velocity in a motion?

To find the constant angular acceleration from the given angular velocity, you can follow the steps mentioned in the article Finding constant angular acceleration in motion. First, determine the final angular velocity and initial angular velocity. Then, calculate the change in angular velocity and the change in time. Finally, divide the change in angular velocity by the change in time to obtain the constant angular acceleration. This method helps in quantifying the change in angular velocity over a specific period of time, enabling a deeper understanding of the motion’s behavior.

Numerical Problems on how to find angular acceleration from angular velocity

Problem 1:

An object is rotating with an angular velocity of 4 rad/s. It accelerates uniformly at a rate of 2 rad/s^2 for a time of 5 seconds. Find the final angular velocity of the object.

Solution:
Given:
Initial angular velocity, $omega_{i} = 4$ rad/s
Angular acceleration, $alpha = 2$ rad/s $^2$
Time, $t = 5$ s

The final angular velocity can be calculated using the formula:
$[omega_{f} = omega_{i} + alpha t]$

Substituting the given values, we have:
$[omega_{f} = 4 + 2 times 5 = 14 text{ rad/s}]$

Therefore, the final angular velocity of the object is 14 rad/s.

Problem 2:

A wheel starts from rest and rotates with an angular acceleration of 3 rad/s^2. If it rotates for a time of 10 seconds, find the final angular velocity of the wheel.

Solution:
Given:
Initial angular velocity, $omega_{i} = 0$ rad/s (as the wheel starts from rest)
Angular acceleration, $alpha = 3$ rad/s $^2$
Time, $t = 10$ s

The final angular velocity can be calculated using the formula:
$[omega_{f} = omega_{i} + alpha t]$

Substituting the given values, we have:
$[omega_{f} = 0 + 3 times 10 = 30 text{ rad/s}]$

Therefore, the final angular velocity of the wheel is 30 rad/s.

Problem 3:

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

angular acceleration from angular velocity 2

A fan blade initially rotates with an angular velocity of 8 rad/s. It decelerates uniformly at a rate of 4 rad/s^2 for a time of 2 seconds. Find the final angular velocity of the fan blade.

Solution:
Given:
Initial angular velocity, $omega_{i} = 8$ rad/s
Angular acceleration, $alpha = -4$ rad/s $^2$ (negative sign indicates deceleration)
Time, $t = 2$ s

The final angular velocity can be calculated using the formula:
$[omega_{f} = omega_{i} + alpha t]$

Substituting the given values, we have:
$[omega_{f} = 8 - 4 times 2 = 0 text{ rad/s}]$

Therefore, the final angular velocity of the fan blade is 0 rad/s.

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AKSHITA MAPARI

Hi, I’m Akshita Mapari. I have done M.Sc. in Physics. I have worked on projects like Numerical modeling of winds and waves during cyclone, Physics of toys and mechanized thrill machines in amusement park based on Classical Mechanics. I have pursued a course on Arduino and have accomplished some mini projects on Arduino UNO. I always like to explore new zones in the field of science. I personally believe that learning is more enthusiastic when learnt with creativity. Apart from this, I like to read, travel, strumming on guitar, identifying rocks and strata, photography and playing chess.

How to Calculate Angular Acceleration from Angular Velocity

The Mathematical Formula

Step-by-Step Process

Worked out Example

Special Cases in Finding Angular Acceleration

How to Find Angular Acceleration without Time

How to Find Angular Acceleration from Angular Velocity and Radius

How to Find Angular Acceleration from Angular Velocity and Time

Related Concepts

How to Find Tangential Acceleration from Angular Velocity

How to Find Linear Acceleration from Angular Velocity

How to Calculate Centripetal Acceleration from Angular Velocity

How can you find the constant angular acceleration from the given angular velocity in a motion?

Numerical Problems on how to find angular acceleration from angular velocity

Problem 1:

Problem 2:

Problem 3:

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