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Concept# Angular acceleration

Summary

In physics, angular acceleration (symbol α, alpha) is the time rate of change of angular velocity. Following the two types of angular velocity, spin angular velocity and orbital angular velocity, the respective types of angular acceleration are: spin angular acceleration, involving a rigid body about an axis of rotation intersecting the body's centroid; and orbital angular acceleration, involving a point particle and an external axis.
Angular acceleration has physical dimensions of angle per time squared, measured in SI units of radians per second squared (rads-2). In two dimensions, angular acceleration is a pseudoscalar whose sign is taken to be positive if the angular speed increases counterclockwise or decreases clockwise, and is taken to be negative if the angular speed increases clockwise or decreases counterclockwise. In three dimensions, angular acceleration is a pseudovector.
For rigid bodies, angular acceleration must be caused by a net external torque. However, this is not so for non-rigid bodies: For example, a figure skater can speed up her rotation (thereby obtaining an angular acceleration) simply by contracting her arms and legs inwards, which involves no external torque.
In two dimensions, the orbital angular acceleration is the rate at which the two-dimensional orbital angular velocity of the particle about the origin changes. The instantaneous angular velocity ω at any point in time is given by
where is the distance from the origin and is the cross-radial component of the instantaneous velocity (i.e. the component perpendicular to the position vector), which by convention is positive for counter-clockwise motion and negative for clockwise motion.
Therefore, the instantaneous angular acceleration α of the particle is given by
Expanding the right-hand-side using the product rule from differential calculus, this becomes
In the special case where the particle undergoes circular motion about the origin, becomes just the tangential acceleration , and vanishes (since the distance from the origin stays constant), so the above equation simplifies to
In two dimensions, angular acceleration is a number with plus or minus sign indicating orientation, but not pointing in a direction.

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