Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared (rad/s2), and is usually denoted by the Greek letter alpha (<math>{\alpha}\,</math>).
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Mathematical definition
The angular acceleration can be defined as either:
- <math>{\alpha} = \frac{d{\omega}}{dt} = \frac{d^2{\theta}}{dt^2}</math> , or
- <math>{\alpha} = \frac{\mathbf{a}_{T}}{r}</math> ,
where <math>{\omega}</math> is the angular velocity, <math>\mathbf{a}_{T}</math> is the linear tangential acceleration, and r is the radius of curvature.
Equations of motion
For rotational motion, Newton's second law can be adapted to describe the relation between torque and angular acceleration:
- <math>{\tau} = I\ {\alpha}</math> ,
where <math>{\tau}</math> is the total torque exerted on the body, and <math>I</math> is the mass moment of inertia of the body.
Constant acceleration
For all constant values of the torque, <math>{\tau}</math>, of an object, the angular acceleration will also be constant. For this special case of constant angular acceleration, the above equation will produce a definitive, singular value for the angular acceleration:
- <math>{\alpha} = \frac{\tau}{I}</math> .
Non-constant acceleration
For any non-constant torque, the angular acceleration of an object will change with time. The equation becomes a differential equation instead of a singular value. This differential equation is known as the equation of motion of the system and can completely describe the motion of the object.
