What is angular displacement?
Imagine an object moving along a circular path, an angle forms along the radius, and this angle is the angular displacement of the object. It means that the body or object is in rotational motion. It is a vector quantity as it has a magnitude and a direction.
Angular displacement formula
We have used various angular displacement equations to formulate our calculator. Let's take a look at all three of them one by one.
- Angular displacement from the radius of the circular path This method uses the radius of the circular path r r and the distance covered along the circular path s s. As a result, you get angular displacement. The formula looks like this: θ = s / r θ=s/r
- Angular displacement from angular velocity Angular velocity is the rate of change of angular displacement. If we shuffle this formula, we can quickly determine the angular displacement from the angular velocity. θ = ω × t θ=ω×t
- Angular displacement from angular acceleration The most common method used to determine angular displacement is through angular acceleration. This formula uses angular velocity, angular acceleration, and time to estimate the angular displacement of the object. θ = ( ω × t ) + ( 1 / 2 × α × t 2 ) θ=(ω×t)+(1/2×α×t 2 )
where:
- θ – Angular displacement;
- s – Distance;
- r – Radius of the circular path;
- ω – Angular velocity;
- t – Time; and
- α – Angular acceleration.
Example
Calculating Angular Displacement
Angular displacement is the angle through which an object rotates or moves about a fixed point or axis in a specific time period. It is a vector quantity, meaning it has both magnitude and direction. The goal of calculating angular displacement is to determine how much an object has rotated over time.
The general approach to calculating angular displacement includes:
- Identifying the initial and final angular velocities of the object.
- Knowing the time taken for the change in angular velocity.
- Applying the formula for angular displacement to calculate the result.
Angular Displacement Formula
The general formula for angular displacement is:
\[ \theta = \frac{{(\omega_2 + \omega_1)}}{{2}} \times t \]Where:
- \( \omega_1 \) is the initial angular velocity of the object (in radians per second, rad/s).
- \( \omega_2 \) is the final angular velocity of the object (in radians per second, rad/s).
- t is the time it takes for the change in angular velocity (in seconds, s).
Example:
If an object accelerates from \( \omega_1 = 2 \, \text{rad/s} \) to \( \omega_2 = 8 \, \text{rad/s} \) in 4 seconds, the angular displacement is:
- Step 1: Add the initial and final angular velocities: \( \omega_2 + \omega_1 = 8 + 2 = 10 \, \text{rad/s} \).
- Step 2: Multiply by the time: \( \theta = \frac{{10}}{{2}} \times 4 = 20 \, \text{rad} \).
Angular Displacement with Changing Direction
Angular displacement can also occur when an object changes direction. For example, when an object rotates in a circle, it experiences angular displacement even if its speed remains constant.
Example:
If a wheel spins at a constant rate of \( 4 \, \text{rad/s} \) for 5 seconds, the angular displacement is:
- Step 1: Multiply the angular velocity by the time: \( \theta = 4 \, \text{rad/s} \times 5 = 20 \, \text{rad} \).
Real-life Applications of Angular Displacement
Calculating angular displacement has many practical applications, such as:
- Determining how much a wheel or gear has rotated over time (e.g., in mechanical engineering).
- Calculating the total rotation of an object in motion (e.g., measuring the number of rotations of a motor's shaft).
- Understanding rotational motion in physics experiments (e.g., determining the angular displacement of an object in circular motion).
Common Units of Angular Displacement
SI Unit: The standard unit of angular displacement is the radian (rad).
Angular displacement can also be expressed in degrees (°), but the radian is the preferred unit in most calculations.
Common Operations with Angular Displacement
Uniform Angular Acceleration: When angular acceleration is constant over time (e.g., an object rotating at a constant rate of change of velocity).
Variable Angular Acceleration: When the angular velocity changes at a non-constant rate (e.g., a rotating object with increasing or decreasing speed).
Negative Angular Displacement: This occurs when the object rotates in the opposite direction (e.g., a wheel rotating counterclockwise after rotating clockwise).
| Problem Type | Description | Steps to Solve | Example |
|---|---|---|---|
| Calculating Angular Displacement from Angular Velocity | Finding the angular displacement when given initial and final angular velocities and time. |
|
For an object that accelerates from \( \omega_1 = 2 \, \text{rad/s} \) to \( \omega_2 = 8 \, \text{rad/s} \) in 4 seconds, the angular displacement is \( \theta = \frac{{8 - 2}}{{2}} \times 4 = 12 \, \text{rad} \). |
| Calculating Angular Displacement from Tangential Acceleration | Finding angular displacement when tangential acceleration is known. |
|
If a point on a rotating wheel has a tangential acceleration of \( 3 \, \text{m/s}^2 \) and the radius is \( 1 \, \text{m} \), the angular displacement in 2 seconds is \( \theta = \frac{{3}}{{1}} \times 2 = 6 \, \text{rad} \). |
| Angular Displacement with Constant Angular Acceleration | Finding angular displacement when angular acceleration is constant over time. |
|
If a rotating object accelerates from \( \omega_1 = 0 \, \text{rad/s} \) to \( \omega_2 = 10 \, \text{rad/s} \) in 5 seconds, the angular displacement is \( \theta = \frac{{10 + 0}}{{2}} \times 5 = 25 \, \text{rad} \). |
| Real-life Applications | Applying angular displacement to solve practical problems in rotational motion. |
|
If a wheel spins at \( 4 \, \text{rad/s} \) for 3 seconds, the angular displacement is \( \theta = 4 \times 3 = 12 \, \text{rad} \). |
