Sums on angular displacement when angular velocity is varying
Sums on Angular Displacement When Angular Velocity is Varying
Angular displacement, angular velocity, and angular acceleration are key concepts in the study of rotational motion. When dealing with problems where the angular velocity is not constant, it is important to understand how to calculate the angular displacement.
Angular Displacement, Velocity, and Acceleration
Before diving into varying angular velocity, let's define the basic terms:
- Angular Displacement ($\theta$): The angle through which a point or line has been rotated in a specified sense about a specified axis. It is measured in radians.
- Angular Velocity ($\omega$): The rate of change of angular displacement and is measured in radians per second (rad/s).
- Angular Acceleration ($\alpha$): The rate of change of angular velocity, measured in radians per second squared (rad/s²).
When Angular Velocity is Constant
In cases where angular velocity is constant, the angular displacement can be calculated using the formula:
[ \theta = \omega t ]
where $\theta$ is the angular displacement in radians, $\omega$ is the constant angular velocity in rad/s, and $t$ is the time in seconds.
When Angular Velocity is Varying
When angular velocity is not constant, it usually means there is an angular acceleration involved. The relationship between angular displacement, initial angular velocity ($\omega_0$), angular acceleration ($\alpha$), and time ($t$) is given by the equation:
[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 ]
This equation is analogous to the linear motion equation $s = ut + \frac{1}{2} at^2$, where $s$ is the displacement, $u$ is the initial velocity, and $a$ is the acceleration.
Table of Differences and Important Points
| Linear Motion | Rotational Motion | Description |
|---|---|---|
| Displacement (s) | Angular Displacement ($\theta$) | In rotational motion, we consider the angle through which an object rotates. |
| Velocity (v) | Angular Velocity ($\omega$) | Angular velocity is the rate of change of angular displacement. |
| Acceleration (a) | Angular Acceleration ($\alpha$) | Angular acceleration is the rate of change of angular velocity. |
| $s = ut + \frac{1}{2} at^2$ | $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$ | The equations of motion for linear and rotational motion are analogous. |
Formulas for Varying Angular Velocity
When angular velocity varies, we can use the following kinematic equations for rotational motion:
- $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$
- $\omega = \omega_0 + \alpha t$
- $\omega^2 = \omega_0^2 + 2\alpha\theta$
These equations are similar to the equations of motion for linear dynamics but are applied to rotational motion.
Examples
Example 1: Constant Angular Acceleration
A wheel starts from rest and has an angular acceleration of 2 rad/s². Calculate the angular displacement after 3 seconds.
Solution:
Given:
- Initial angular velocity, $\omega_0 = 0$ rad/s (since it starts from rest)
- Angular acceleration, $\alpha = 2$ rad/s²
- Time, $t = 3$ s
Using the formula for angular displacement:
[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 ]
[ \theta = 0 \cdot 3 + \frac{1}{2} \cdot 2 \cdot 3^2 ]
[ \theta = 0 + \frac{1}{2} \cdot 2 \cdot 9 ]
[ \theta = 9 \text{ radians} ]
Example 2: Finding Final Angular Velocity
A disk rotates with an initial angular velocity of 10 rad/s and accelerates uniformly at a rate of 5 rad/s². Find the final angular velocity after 4 seconds.
Solution:
Given:
- Initial angular velocity, $\omega_0 = 10$ rad/s
- Angular acceleration, $\alpha = 5$ rad/s²
- Time, $t = 4$ s
Using the formula for final angular velocity:
[ \omega = \omega_0 + \alpha t ]
[ \omega = 10 + 5 \cdot 4 ]
[ \omega = 10 + 20 ]
[ \omega = 30 \text{ rad/s} ]
These examples illustrate how to use the kinematic equations for rotational motion to solve problems involving varying angular velocity. Understanding these concepts is crucial for solving physics problems related to rotational dynamics.