Angular acceleration

Radians per second squared
Unit system	SI derived unit
Unit of	Angular acceleration
Symbol	rad/s2, rad⋅s−2

Angular acceleration is the rate of change of angular velocity. In SI units, it is measured in radians per second squared (rad/s²), and is usually denoted by the Greek letter alpha (α).

Mathematical definition

The angular acceleration can be defined as either:

${\alpha }={\frac {d\omega }{dt}}={\frac {d^{2}{\theta }}{dt^{2}}}$ , or

${\alpha }={\frac {a_{T}}{r}}$ ,

where ${\omega }$ is the angular velocity, $a_{T}$ is the linear tangential acceleration, and $r$, (usually defined as the radius of the circular path of which a point moving along), is the distance from the origin of the coordinate system that defines $\theta$ and $\omega$ to the point of interest.

Equations of motion

For two-dimensional rotational motion (constant ${\hat {L}}$), Newton's second law can be adapted to describe the relation between torque and angular acceleration:

${\tau }=I\ {\alpha }$ ,

where ${\tau }$ is the total torque exerted on the body, and $I$ is the mass moment of inertia of the body.

Constant acceleration

For all constant values of the torque, ${\tau }$, 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, constant value for the angular acceleration:

${\alpha }={\frac {\tau }{I}}.$

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 constant value. This differential equation is known as the equation of motion of the system and can completely describe the motion of the object. It is also the best way to calculate the angular velocity.

Relationship with Angular Momentum

Angular acceleration can be obtained from angular momentum using the relation ,

${\alpha }=I^{-1}(L\times \omega +{\frac {dL}{dt}})$

Above relationship indicates that even when there is no change in angular momentum (i.e. no torques are being applied), the angular acceleration can still be non-zero. In fact, this will happen whenever the angular momentum and angular velocity point in different directions (i.e. rotational velocity axis is not the axis of symmetry).

See also

• Angular momentum
• Angular speed
• Angular velocity
• Rotation
• Spin