Understanding how to determine the final angular speed of a rotating object is crucial in various fields, such as mechanics and engineering. This article will guide you on how to find the final angular speed using a formula and will provide two examples to illustrate the process.
Formula for Final Angular Speed
To find the final angular speed (\(w_2\)), use the following formula derived from the angular acceleration formula:
\[ w_2 = w_1 + \alpha \cdot t \]
Where:
- \(w_2\) is the final angular speed in radians per second (\(\text{rad/s}\)).
- \(w_1\) is the initial angular speed in radians per second (\(\text{rad/s}\)).
- \(\alpha\) is the angular acceleration in radians per second squared (\(\text{rad/s}^2\)).
- \(t\) is the time interval in seconds (\(\text{s}\)).
Example 1: Positive Angular Acceleration
In this example, we will calculate the final angular speed when the angular acceleration is positive.
Given:
- Initial angular speed \(w_1 = 10 \, \text{rad/s}\)
- Angular acceleration \(\alpha = 4 \, \text{rad/s}^2\)
- Time \(t = 5 \, \text{s}\)
Step-by-Step Calculation:
Step 1: Identify the Given Values
Given:
- \(w_1 = 10 \, \text{rad/s}\)
- \(\alpha = 4 \, \text{rad/s}^2\)
- \(t = 5 \, \text{s}\)
Step 2: Substitute the Values into the Formula for Final Angular Speed
Using the formula:
\[ w_2 = w_1 + \alpha \cdot t \]
Substitute \(w_1 = 10 \, \text{rad/s}\), \(\alpha = 4 \, \text{rad/s}^2\), and \(t = 5 \, \text{s}\):
\[ w_2 = 10 + 4 \cdot 5 \]
Step 3: Calculate the Final Angular Speed
\[ w_2 = 10 + 20 = 30 \, \text{rad/s} \]
Final Value
The final angular speed is \(30 \, \text{rad/s}\).
Example 2: Negative Angular Acceleration
Now, let's calculate the final angular speed when the angular acceleration is negative.
Given:
- Initial angular speed \(w_1 = 15 \, \text{rad/s}\)
- Angular acceleration \(\alpha = -3 \, \text{rad/s}^2\)
- Time \(t = 6 \, \text{s}\)
Step-by-Step Calculation:
Step 1: Identify the Given Values
Given:
- \(w_1 = 15 \, \text{rad/s}\)
- \(\alpha = -3 \, \text{rad/s}^2\)
- \(t = 6 \, \text{s}\)
Step 2: Substitute the Values into the Formula for Final Angular Speed
Using the formula:
\[ w_2 = w_1 + \alpha \cdot t \]
Substitute \(w_1 = 15 \, \text{rad/s}\), \(\alpha = -3 \, \text{rad/s}^2\), and \(t = 6 \, \text{s}\):
\[ w_2 = 15 + (-3) \cdot 6 \]
Step 3: Calculate the Final Angular Speed
\[ w_2 = 15 - 18 = -3 \, \text{rad/s} \]
Final Value
The final angular speed is \(-3 \, \text{rad/s}\).
Summary
To determine the final angular speed (\(w_2\)) of a rotating object when the initial angular speed (\(w_1\)), angular acceleration (\(\alpha\)), and time interval (\(t\)) are known, use the formula:
\[ w_2 = w_1 + \alpha \cdot t \]
Whether the angular acceleration is positive or negative, this formula allows you to calculate the final angular speed. This calculation is essential for applications in rotational dynamics, helping to predict how quickly an object will spin after a certain period under constant acceleration.
These examples demonstrate how to apply the formula in various scenarios, providing a clear understanding of how to compute the final angular speed based on given parameters.
