How To Find Constant Angular Acceleration: Problems And Examples

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.

Numerical Problems on how to find constant angular acceleration

Problem 1:

A wheel starts from rest and accelerates with a constant angular acceleration of 2 rad/s^2 for a time interval of 5 seconds. Find the angular velocity of the wheel at the end of the time interval.

Solution:

Given:
Initial angular velocity, $\omega_i = 0$ rad/s
Angular acceleration, $\alpha = 2$ rad/s^2
Time, $t = 5$ s

Using the formula for angular velocity with constant angular acceleration:

$[\omega_f = \omega_i + \alpha \cdot t]$

Substituting the given values:

$[\omega_f = 0 + 2 \cdot 5]$

Simplifying:

$[\omega_f = 10 \text{ rad/s}]$

Therefore, the angular velocity of the wheel at the end of the time interval is 10 rad/s.

Problem 2:

A spinning top starts from rest and accelerates with a constant angular acceleration of 1.5 rad/s^2. If it takes 8 seconds for the top to reach a certain angular velocity, find the final angular velocity.

Solution:

Given:
Initial angular velocity, $\omega_i = 0$ rad/s
Angular acceleration, $\alpha = 1.5$ rad/s^2
Time, $t = 8$ s

Using the formula for angular velocity with constant angular acceleration:

$[\omega_f = \omega_i + \alpha \cdot t]$

Substituting the given values:

$[\omega_f = 0 + 1.5 \cdot 8]$

Simplifying:

$[\omega_f = 12 \text{ rad/s}]$

Therefore, the final angular velocity of the spinning top is 12 rad/s.

Problem 3:

A flywheel starts from rest and accelerates with a constant angular acceleration of 4 rad/s^2. If the angular displacement covered by the flywheel in a certain time interval is 10 radians, find the time interval.

Solution:

Given:
Initial angular velocity, $\omega_i = 0$ rad/s
Angular acceleration, $\alpha = 4$ rad/s^2
Angular displacement, $\theta = 10$ radians

Using the formula for angular displacement with constant angular acceleration:

$[\theta = \omega_i \cdot t + \frac{1}{2} \alpha \cdot t^2]$

Rearranging the equation:

$[\frac{1}{2} \alpha \cdot t^2 + \omega_i \cdot t - \theta = 0]$

Substituting the given values:

$[\frac{1}{2} \cdot 4 \cdot t^2 + 0 \cdot t - 10 = 0]$

Simplifying:

$[2t^2 - 10 = 0]$

Solving the quadratic equation, we find $t = \pm \sqrt{5}$

Since time cannot be negative, the time interval is:

$[t = \sqrt{5} \text{ s}]$

Therefore, the time interval for the flywheel to cover an angular displacement of 10 radians is approximately $\sqrt{5}$ seconds.

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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 Constant Angular Acceleration

Constant Angular Acceleration Formula

Steps to Calculate Constant Angular Acceleration

Worked Out Example: Calculating Constant Angular Acceleration

How to Determine Angular Acceleration with Angular Velocity

Relationship between Angular Velocity and Angular Acceleration

Steps to Determine Angular Acceleration using Angular Velocity

Worked Out Example: Finding Angular Acceleration with Angular Velocity

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

Numerical Problems on how to find constant angular acceleration

Problem 1:

Problem 2:

Problem 3:

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