In uniform rotational motion, the angular acceleration is constant.
Rotational motion describes the movement of an object around an axis or center point. While many quantities can change during rotational motion (like angular velocity or angular displacement), certain aspects remain constant under specific conditions.
Understanding Uniform Rotational Motion
Uniform rotational motion is a specific type of rotation where the object rotates at a steady rate. This means its angular velocity is changing by the same amount each second.
According to the provided reference, "In uniform rotational motion, the angular acceleration is constant so it can be pulled out of the integral, yielding two definite integrals: α ∫ t 0 t d t ′ = ∫ ω 0 ω f d ω ." This mathematical representation shows how the constant angular acceleration (α) simplifies the relationship between angular velocity (ω) and time (t).
Key Constant in Uniform Rotation
The primary constant in uniform rotational motion is the angular acceleration (α). This value tells us how quickly the angular velocity is changing. If angular acceleration is constant, it can be zero (in the case of constant angular velocity, which is sometimes also referred to as a type of uniform motion, though the reference specifically highlights constant angular acceleration), or it can be a non-zero constant value.
Rotational Motion Variables
To better understand what remains constant, let's look at the key variables in rotational motion:
| Variable | Symbol | Description | Unit (SI) |
|---|---|---|---|
| Angular Displacement | θ | Angle through which an object rotates | radians (rad) |
| Angular Velocity | ω | Rate of change of angular displacement | rad/s |
| Angular Acceleration | α | Rate of change of angular velocity | rad/s² |
| Time | t | Duration of motion | seconds (s) |
In uniform rotational motion as described by the reference, while angular displacement and angular velocity may change over time, the angular acceleration (α) remains the same.
Implications of Constant Angular Acceleration
A constant angular acceleration simplifies the equations used to describe the motion, much like constant linear acceleration does in translational motion. It allows us to use straightforward kinematic equations to relate angular displacement, angular velocity, angular acceleration, and time.
For example, if you know the initial angular velocity (ω₀) and the constant angular acceleration (α), you can find the final angular velocity (ωf) after a time (t) using:
ωf = ω₀ + αt
This equation is only valid when the angular acceleration is constant, highlighting the importance of this condition.
Practical Insights
Consider a merry-go-round starting from rest and speeding up smoothly. If the motor applies a constant torque, the merry-go-round experiences a constant angular acceleration (assuming constant moment of inertia and neglecting friction). During this phase of motion, the angular acceleration is constant, allowing engineers to predict its angular velocity and position at any given time.
In summary, while rotational motion can involve many changing quantities, in the specific case of uniform rotational motion, the angular acceleration is the quantity that remains constant.
