In physics, angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity, there are naturally also two types of angular acceleration, called spin angular acceleration and orbital angular acceleration respectively. Spin angular acceleration refers to the angular acceleration of a rigid body about its centre of rotation, and orbital angular acceleration refers to the angular acceleration of a point particle about a fixed origin. Angular acceleration is measured in units of angle per unit time squared, and is usually represented by the symbol alpha. By convention, positive angular acceleration indicates an increase in counter-clockwise angular speed or decrease in clockwise angular speed, while negative angular acceleration indicates an increase in clockwise angular speed or decrease in counterclockwise angular speed. 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 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, 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, 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. The sign is conventionally taken to be positive if the angular speed increases in the counter-clockwise direction or decreases in the clockwise direction, and the sign is taken negative if the angular speed increases in the clockwise direction or decreases in the counter-clockwise direction. Angular acceleration then may be termed a pseudoscalar, a numerical quantity which changes sign under a parity inversion, such as inverting one axis or switching the two axes.
Particle in three dimensions
In three dimensions, the orbital angular acceleration is the rate at which three-dimensional orbital angular velocity vector changes with time. The instantaneous angular velocity vector is defined by Therefore, the orbital angular acceleration vector is given by Expanding this derivative using the product rule for cross-products and the ordinary quotient rule, one gets: Since is just the radial component of the velocity vector, and is just, the second term may be rewritten more compactly as. In the case where the distance of the particle from the origin does not change with time, the second term vanishes and the above formula simplifies to From the above equation, one can recover the cross-radial acceleration in this special case as: Unlike in two dimensions, the angular acceleration in three dimensions need not be associated with a change in angular speed: If the particle's trajectory "twists" in space such that the instantaneous plane of angular velocity continuously changes with time, then even if the angular speed is constant, there will still be a nonzero angular acceleration because the direction of the angular velocity vector changes with time. This cannot not happen in two dimensions because the position vector is restricted to a fixed plane so that any change in angular velocity must be through a change in its magnitude. In three dimensions, angular acceleration is what is called a pseudovector. It has three components which transform under rotations the same way as a vector does, but which under reflections transforms differently than a vector.
Relation to Torque
The net torque on a point particle is given by where F is the net force on the particle. By Newton's Second Law, this may be rewritten as where is the mass of the particle. Dividing both sides by the moment of inertia of the particle about the origin, this becomes But from the previous section, it was derived that where is the orbital angular acceleration of the particle and is the orbital angular velocity of the particle. Therefore, it follows that The above relationship is the rotational analog of Newton's Second Law; it connects the net applied torque on a point particle to the induced angular acceleration of the particle. Unlike the relationship between force and acceleration, the angular acceleration vector is not necessarily parallel or directly proportional to the torque vector, and does not depend solely on the net torque and moment of inertia. However, in the special case where the distance to the origin does not change with time, the second term in the above equation vanishes and the above equation simplifies to
