Angular acceleration is a measure of how quickly the angular velocity of a rotating object changes with time. It is a fundamental concept in rotational motion and is essential for understanding the dynamics of rotating systems. This article will show you how to calculate angular acceleration with detailed steps and examples.
Formula for Angular Acceleration
The formula for calculating angular acceleration (\(\alpha\)) is:
\[ \alpha = \dfrac{w_2 - w_1}{t} \]
Where:
- \(\alpha\) is the angular acceleration.
- \(w_1\) is the initial angular speed in radians per second (\(\text{rad/s}\)).
- \(w_2\) is the final angular speed in radians per second (\(\text{rad/s}\)).
- \(t\) is the time interval in seconds (\(\text{s}\)).
Example 1: Initial Angular Speed is Zero
Let’s start with an example where the initial angular speed (\(w_1\)) is zero.
Given:
- Initial angular speed \(w_1 = 0 \, \text{rad/s}\)
- Final angular speed \(w_2 = 10 \, \text{rad/s}\)
- Time \(t = 5 \, \text{s}\)
Step-by-Step Calculation:
Step 1: Identify the Given Values
Given:
- \(w_1 = 0 \, \text{rad/s}\)
- \(w_2 = 10 \, \text{rad/s}\)
- \(t = 5 \, \text{s}\)
Step 2: Substitute the Values into the Angular Acceleration Formula
Using the formula:
\[ \alpha = \dfrac{w_2 - w_1}{t} \]
Substitute \(w_1 = 0 \, \text{rad/s}\), \(w_2 = 10 \, \text{rad/s}\), and \(t = 5 \, \text{s}\):
\[ \alpha = \dfrac{10 - 0}{5} \]
Step 3: Calculate the Angular Acceleration
\[ \alpha = \dfrac{10}{5} = 2 \, \text{rad/s}^2 \]
Final Value
The angular acceleration is \(2 \, \text{rad/s}^2\).
Example 2: Initial Angular Speed is Non-Zero
Let’s consider another example where the initial angular speed is non-zero.
Given:
- Initial angular speed \(w_1 = 4 \, \text{rad/s}\)
- Final angular speed \(w_2 = 14 \, \text{rad/s}\)
- Time \(t = 4 \, \text{s}\)
Step-by-Step Calculation:
Step 1: Identify the Given Values
Given:
- \(w_1 = 4 \, \text{rad/s}\)
- \(w_2 = 14 \, \text{rad/s}\)
- \(t = 4 \, \text{s}\)
Step 2: Substitute the Values into the Angular Acceleration Formula
Using the formula:
\[ \alpha = \dfrac{w_2 - w_1}{t} \]
Substitute \(w_1 = 4 \, \text{rad/s}\), \(w_2 = 14 \, \text{rad/s}\), and \(t = 4 \, \text{s}\):
\[ \alpha = \dfrac{14 - 4}{4} \]
Step 3: Calculate the Angular Acceleration
\[ \alpha = \dfrac{10}{4} = 2.5 \, \text{rad/s}^2 \]
Final Value
The angular acceleration is \(2.5 \, \text{rad/s}^2\).
Example 3: Final Angular Speed is Slower Than Initial Angular Speed
Now, let’s consider a scenario where the final angular speed is slower than the initial angular speed.
Given:
- Initial angular speed \(w_1 = 20 \, \text{rad/s}\)
- Final angular speed \(w_2 = 8 \, \text{rad/s}\)
- Time \(t = 6 \, \text{s}\)
Step-by-Step Calculation:
Step 1: Identify the Given Values
Given:
- \(w_1 = 20 \, \text{rad/s}\)
- \(w_2 = 8 \, \text{rad/s}\)
- \(t = 6 \, \text{s}\)
Step 2: Substitute the Values into the Angular Acceleration Formula
Using the formula:
\[ \alpha = \dfrac{w_2 - w_1}{t} \]
Substitute \(w_1 = 20 \, \text{rad/s}\), \(w_2 = 8 \, \text{rad/s}\), and \(t = 6 \, \text{s}\):
\[ \alpha = \dfrac{8 - 20}{6} \]
Step 3: Calculate the Angular Acceleration
\[ \alpha = \dfrac{-12}{6} = -2 \, \text{rad/s}^2 \]
Final Value
The angular acceleration is \(-2 \, \text{rad/s}^2\).
Summary
To calculate angular acceleration (\(\alpha\)) when the initial and final angular speeds, along with time, are known, use the formula:
\[ \alpha = \dfrac{w_2 - w_1}{t} \]
Whether the initial angular speed is zero, non-zero, or the final angular speed is slower than the initial angular speed, this formula helps determine the rate at which angular velocity changes over time.
