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 Find Angular Acceleration From Angular Velocity: Problem And Examples

January 21, 2024January 17, 2022 by AKSHITA MAPARI

angular acceleration from angular velocity 0

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.

Also Read:

11 Angular Acceleration Examples: Detailed Insight And Facts

January 21, 2024January 17, 2022 by AKSHITA MAPARI

In this article, we will see various angular acceleration examples with detailed insight.

An object exhibiting circular motion associates the angular acceleration. The following is the list of examples of angular acceleration:-

Electric Motor

The electric motor runs on the principle of electromagnetism. When supplied with the electric current, the coil in the motor produces the magnetic field that experiences the torque and the motor rotates. It helps to convert electric energy into mechanical energy. The speed of the motor is calculated as the rotation per minute (rpm).

Merry-go-round

A person sitting on the merry-go-round experiences the centripetal force acting towards the point at a center, and the centrifugal force outward. Both the forces cancel out, which keeps the body intact on the seat and in the direction of motion.

angular acceleration examples — **Merry-go-round;**
**Image Credit: Pixabay**

There is a change in the angular velocity of the merry-go-round while speeding up and slowing down the wheel. The direction of the radial velocity of a person keeps on changing which is proportional to the angular velocity and the distance of a chair on which the person is seated from the center of a merry-go-round.

Ferries wheel

The ferries wheel accelerates at a constant speed, and its speed varies only at the start of the journey and at the end to slow down. Since, there is a change in the angular velocity of a ferries wheel, while speeding up or down it is associated with the angular acceleration; otherwise, it has a constant angular acceleration.

ferris wheel g4a9f91883 640 — **Ferrieswheel;**
**Image Credit: Pixabay**

We can write a kinematics equation for the angular velocity of the ferries wheel as

Integrating the above equation,

Squaring eqn(1) on both the sides

From eqn(2),

Using this in the above eqn, we have third kinematic equation.

Mixer

Due to the rotation of the shaft attached to the blades used in a mixer, the blades rotate to grind the mixer. The speed of the motor is regulated by the power supplied to the machine that increases or decreases the rotations per second of the motor couplers. As the angular velocity of the motor coupler varies, the rate of angular acceleration varies accordingly.

Grinder

A shaft on the machine rotates on supplying the electric current that results in the rotors attached to the shafts to accelerate. One rotation of a rotor will displace the outer vessel which is filled with the mixture to a length of the circumference of the rotor.

The angular velocity of the grinder varies on changing the power supply while starting the machine and on putting off from the power supply.

Gravitron

The angular velocity of the gravitron rises gradually and constantly until the centrifugal force acting on the body reaches the epic point at which, even after removing the plate beneath the foot of a person standing in the gravitron does not fall down due to gravity.

This is an exciting thing to experience at the amusement park. The speed of a gravitron varies constantly and hence there is an angular acceleration of the gravitron. The centrifugal force is responsible for a body to remain attached to the wall of the gravitron and prevent a person from falling down underneath.

Wheel of a Car

The angular velocity of a wheel depends upon the speed of a car. The angular acceleration of a wheel is equal to the change in the angular velocity of a wheel.

automobile g4cc147d35 640 — **A car; Image Credit: Pixabay**

The distance covered by one revolution of a wheel of a car is equal to the circumference of a wheel. The greater the diameter of a wheel less will be its angular velocity and hence less will be the angular acceleration by the wheel.

Fans

On putting on the power to the fan, the propellers initially rotate because of the energy stored in the capacitor of the fan, and then gradually take the speed as the motor of the fan experiences the torque due to the electromagnetic field.

ceiling fan g025e9aa3f 640 — **Ceiling Fan,**
**Image Credit: Pixabay**

The speed of the rotation of a propeller is regulated by the regulator and changes by switching the plug for different power ranges. We see the variations in the angular velocity of the fan, hence is an example of angular acceleration.

Windmills

Windmill converts wind energy to electric energy. For propellers of the windmill to rotate a shaft is connected to the propellers and which is then connected to the motor to increase the number of rotations of a propeller to produce a large amount of energy. The electric energy produced is stored in the generator.

windmill g5f50d7666 640 — **Windmill;**
**Image Credit: Pixabay**

The energy produced depends upon the number of turns of a propeller which is governed by the angular speed that keeps on varying with the speed of the wind. Hence, the acceleration rate of the propellers varies.

Pulley

A pulley is a wheel that is designed for easy displace or lift the objects to minimize the utilization of the muscular force by changing the direction of the force applied.

See the source image — **Pulley; Image Credit: GlobalSpec**

The angular acceleration of a pulley depends upon the angular velocity of a pulley varied with time. The pulley accelerates by stretching the length of the string attached to the pulley.

Pendulum in a Simple Harmonic Motion

A pendulum in a harmonic motion oscillates with angular velocity making an angle between the initial and final highest point which reduces with time. The speed of the pendulum also decreases due to the restoring force and the air drag.

There is a continuous change in the angular velocity of a pendulum which decreases with time; hence the angular acceleration of a pendulum in a simple harmonic motion is negative.

Frequently Asked Questions

Q1. The ferries wheel at a rest starts accelerating with an angular velocity of 4m/s in 8 seconds. What is the angular acceleration of the ferries wheel?

Given:

ω_f= 4m/s

ω_i= 0

t= 8s

ω_f= ω_i + α t

4= 0+ α * 8

α = 4/8 = 0.5 m/s²]

Q2. Consider an object in a rotational motion. The angular displacement of the object is given by the equation θ = 3πt² + πt + 6π + 2 Find the angular velocity and angular acceleration of the object.

Given:

The angular velocity is given by

The angular acceleration of the object is

What is orbital angular acceleration?

An object moving in an orbital motion with focus at the center making an angle θ on displacement.

The change in the angular velocity of the object in an orbital motion due to varying angles with respect to time is called the orbital angular acceleration.

Click to read more on 16+ Change In Acceleration Examples.

Also Read:

How To Find Constant Angular Acceleration: Problems And Examples

January 21, 2024January 15, 2022 by AKSHITA MAPARI

Angular acceleration is a key concept in rotational motion, describing how quickly an object’s angular velocity changes over time. In this blog post, we will explore how to find constant angular acceleration, which occurs when the angular acceleration remains the same throughout the motion. We will discuss the formulas, calculation steps, and provide worked-out examples to help you understand and apply this concept effectively.

How to Calculate Constant Angular Acceleration

Constant Angular Acceleration Formula

To calculate constant angular acceleration, we can use the following formula:

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

Where:
– $\alpha$ represents the constant angular acceleration,
– $\Delta \omega$ is the change in angular velocity, and
– $\Delta t$ is the change in time.

Steps to Calculate Constant Angular Acceleration

Image by Cdang – Wikimedia Commons, Wikimedia Commons, Licensed under CC0.

The following steps outline how to calculate constant angular acceleration:

Determine the initial angular velocity ( $\omega_i$ ) and final angular velocity ( $\omega_f$ ).
Determine the initial time ( $t_i$ ) and final time ( $t_f$ ).
Calculate the change in angular velocity ( $\Delta \omega$ ) by subtracting the initial angular velocity from the final angular velocity: $\Delta \omega = \omega_f - \omega_i$ .
Calculate the change in time ( $\Delta t$ ) by subtracting the initial time from the final time: $\Delta t = t_f - t_i$ .
Use the constant angular acceleration formula mentioned earlier to find the value of angular acceleration ( $\alpha$ ) by dividing the change in angular velocity by the change in time: $\alpha = \frac{{\Delta \omega}}{{\Delta t}}$ .

Worked Out Example: Calculating Constant Angular Acceleration

Let’s consider an example to illustrate how to calculate constant angular acceleration.

Suppose a disc starts from rest and rotates with an angular velocity of 20 rad/s after 5 seconds. We need to find the constant angular acceleration.

Given:
– Initial angular velocity ( $\omega_i$ ) = 0 rad/s
– Final angular velocity ( $\omega_f$ ) = 20 rad/s
– Initial time ( $t_i$ ) = 0 s
– Final time ( $t_f$ ) = 5 s

Step 1: Determine the change in angular velocity:
$\Delta \omega = \omega_f - \omega_i = 20 - 0 = 20 \, \text{rad/s}$

Step 2: Determine the change in time:
$\Delta t = t_f - t_i = 5 - 0 = 5 \, \text{s}$

Step 3: Calculate the constant angular acceleration:
$\alpha = \frac{{\Delta \omega}}{{\Delta t}} = \frac{{20}}{{5}} = 4 \, \text{rad/s}^2$

Therefore, the constant angular acceleration of the disc is 4 rad/s^2.

How to Determine Angular Acceleration with Angular Velocity

Relationship between Angular Velocity and Angular Acceleration

Angular velocity and angular acceleration are closely related. If the angular acceleration is constant, we can determine the angular acceleration using the initial and final angular velocities, along with the time taken.

The relationship between angular velocity ( $\omega$ ), angular acceleration ( $\alpha$ ), and time ( $t$ ) can be described by the equation:

$\omega_f = \omega_i + \alpha t$

Where:
– $\omega_i$ and $\omega_f$ are the initial and final angular velocities, respectively,
– $\alpha$ is the constant angular acceleration, and
– $t$ is the time taken.

Steps to Determine Angular Acceleration using Angular Velocity

Image by Pradana Aumars – Wikimedia Commons, Wikimedia Commons, Licensed under CC0.

To determine the angular acceleration using angular velocity, follow these steps:

Identify the initial angular velocity ( $\omega_i$ ), final angular velocity ( $\omega_f$ ), and time ( $t$ ).
Substitute the given values into the equation $\omega_f = \omega_i + \alpha t$ .
Rearrange the equation to solve for the angular acceleration ( $\alpha$ ): $\alpha = \frac{{\omega_f - \omega_i}}{{t}}$ .

Worked Out Example: Finding Angular Acceleration with Angular Velocity

Let’s work through an example to understand how to find the angular acceleration using angular velocity.

Suppose a wheel starts with an initial angular velocity of 10 rad/s and reaches a final angular velocity of 30 rad/s in 5 seconds. We want to determine the angular acceleration.

Given:
– Initial angular velocity ( $\omega_i$ ) = 10 rad/s
– Final angular velocity ( $\omega_f$ ) = 30 rad/s
– Time ( $t$ ) = 5 s

Step 1: Use the equation $\omega_f = \omega_i + \alpha t$ with the given values:
30 = 10 + $\alpha$ × 5

Step 2: Rearrange the equation to solve for $\alpha$ :
$\alpha = \frac{{\omega_f - \omega_i}}{{t}} = \frac{{30 - 10}}{{5}} = 4 \, \text{rad/s}^2$

Thus, the angular acceleration of the wheel is 4 rad/s^2.

Understanding how to find constant angular acceleration is essential for analyzing rotational motion. By using the constant angular acceleration formula and the relationship between angular velocity and angular acceleration, you can determine the angular acceleration of an object. Remember to follow the steps we discussed and utilize the provided formulas when solving problems involving constant angular acceleration. Practice applying these concepts with different examples, and you’ll soon become proficient in calculating and understanding constant angular acceleration.

How can the concept of constant angular acceleration be applied to finding the angular acceleration of a wheel?

The process of finding the angular acceleration of a wheel involves understanding the concept of constant angular acceleration. By analyzing the angular motion of the wheel and considering factors such as its radius and linear acceleration, it is possible to determine the angular acceleration. For a detailed guide on how to find the angular acceleration of a wheel, you can refer to the article on Finding angular acceleration of a wheel.