Rotational acceleration

In physics, **rotational acceleration** or **angular acceleration** refers to the time rate of change of rotational velocity. As there are two types of rotational velocity, namely spin rotational velocity and orbital rotational velocity, there are naturally also two types of rotational acceleration, called spin rotational acceleration and orbital rotational acceleration respectively. Spin rotational acceleration refers to the rotational acceleration of a rigid body about its centre of rotation, and orbital rotational acceleration refers to the rotational acceleration of a point particle about a fixed origin.

Rotational acceleration is measured in units of angle per unit time squared (which in SI units is radians per second squared), and is usually represented by the symbol alpha (**α**). In two dimensions, rotational 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, rotational acceleration is a pseudovector.

For rigid bodies, rotational acceleration must be caused by a net external rotational force. However, this is not so for non-rigid bodies: For example, a figure skater can speed up her rotation (thereby obtaining an rotational acceleration) simply by contracting her arms and legs inwards, which involves no *external* rotational force.

## Orbital Angular Acceleration of a Point Particle

### Particle in two dimensions

In two dimensions, the orbital rotational acceleration is the rate at which the two-dimensional orbital rotational velocity of the particle about the origin changes. The instantaneous rotational velocity *ω* at any point in time is given by

# ω

v

⊥

r

{\displaystyle \omega = \frac{v\_{\perp}}{r}}

,
where

r

{\displaystyle r}

 is the distance from the origin and

v
⊥

{\displaystyle v\_\perp}

 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 rotational acceleration *α* of the particle is given by

# α

d

d
t

(

v

⊥

r

)

{\displaystyle \alpha = \frac{d}{dt}(\frac{v\_{\perp}}{r})}

.

Expanding the right-hand-side using the product rule from differential calculus, this becomes

# α

1
r

d

v

⊥

d
t

−

v

⊥

r
2

d
r

d
t

{\displaystyle \alpha = \frac{1}{r}\frac{dv\_{\perp}}{dt} - \frac{v\_{\perp}}{r^2}\frac{dr}{dt}}

.
In the special case where the particle undergoes circular motion about the origin,

d

v

⊥

d
t

{\displaystyle \frac{dv\_{\perp}}{dt}}

 becomes just the tangential acceleration

a

⊥

{\displaystyle a\_{\perp}}

, and

d
r

d
t

{\displaystyle \frac{dr}{dt}}

 vanishes (since the distance from the origin stays constant), so the above equation simplifies to

# α

a

⊥

r

{\displaystyle \alpha = \frac{a\_{\perp}}{r}}

.

In two dimensions, rotational 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. Rotational 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 rotational acceleration is the rate at which three-dimensional orbital rotational velocity vector changes with time. The instantaneous rotational velocity vector

ω

{\displaystyle \boldsymbol{\omega}}

 at any point in time is given by

# ω

r

×

v

r
2

{\displaystyle \boldsymbol{\omega}=\frac{\mathbf{r}\times\mathbf{v}}{r^2}}

,
where

r

{\displaystyle \mathbf{r}}

 is the particle's position vector and

v

{\displaystyle \mathbf{v} }

 is its velocity vector.

Therefore, the orbital rotational acceleration is the vector

α

{\displaystyle \boldsymbol{\alpha}}

 defined by

# α

d

d
t

(

r

×

v

r
2

)

{\displaystyle \boldsymbol\alpha = \frac{d}{dt}(\frac{\mathbf r\times\mathbf v}{r^2})}

.
Expanding this derivative using the product rule for cross-products and the ordinary quotient rule, one gets:

α

=

1

r
2

(

r

×

d

v

d
t

d

r

d
t

×

v

)
−

2

r
3

d
r

d
t

(

r

×

v

)

=

1

r
2

(

r

×

a

v

×

v

)
−

2

r
3

d
r

d
t

(

r

×

v

)

=

r

×

a

r
2

−

2

r
3

d
r

d
t

(

r

×

v

)
.

{\displaystyle \begin{align}
\boldsymbol\alpha &= \frac{1}{r^2}(\mathbf r\times \frac{d\mathbf v}{dt} + \frac{d\mathbf r}{dt}\times \mathbf v)-\frac{2}{r^3}\frac{dr}{dt}(\mathbf r\times\mathbf v)\\
\\
&= \frac{1}{r^2}(\mathbf r\times \mathbf a + \mathbf v\times \mathbf v)-\frac{2}{r^3}\frac{dr}{dt}(\mathbf r\times\mathbf v)\\
\\
&= \frac{\mathbf r\times \mathbf a}{r^2}-\frac{2}{r^3}\frac{dr}{dt}(\mathbf r\times\mathbf v).
\end{align}}

Since

r

×

v

{\displaystyle \mathbf r\times\mathbf v}

 is just

r
2

ω

{\displaystyle r^2\boldsymbol{\omega}}

, the second term may be rewritten as

−

2
r

d
r

d
t

ω

{\displaystyle -\frac{2}{r}\frac{dr}{dt}\boldsymbol{\omega}}

. In the case where the distance

r

{\displaystyle r}

 of the particle from the origin does not change with time (which includes circular motion as a subcase), the second term vanishes and the above formula simplifies to

# α

r

×

a

r
2

{\displaystyle \boldsymbol\alpha = \frac{\mathbf r\times \mathbf a}{r^2}}

.
From the above equation, one can recover the cross-radial acceleration in this special case as:

a

⊥

=
α
×

r

{\displaystyle \mathbf{a}\_{\perp} =\boldsymbol{\alpha} \times\mathbf{r}}

.
Unlike in two dimensions, the rotational acceleration in three dimensions need not be associated with a change in angular *speed*: If the particle's position vector "twists" in space such that its instantaneous plane of angular displacement (i.e. the instantaneous plane in which the position vector sweeps out angle) continuously changes with time, then even if the angular speed (i.e. the speed at which the position vector sweeps out angle) is constant, there will still be a nonzero rotational acceleration because the *direction* of the rotational velocity vector continuously 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 rotational velocity must be through a change in its *magnitude*.

The rotational acceleration vector is more properly called a pseudovector: It has three components which transform under rotations in the same way as the Cartesian coordinates of a point do, but which under reflections do not transform like Cartesian coordinates.

### Relation to Rotational force

The net *rotational force* on a point particle is defined to be the pseudovector

# τ

r

×

F

{\displaystyle \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}}

,
where

F

{\displaystyle \mathbf F }

 is the net force on the particle.

Rotational force is the rotational analogue of force: it induces change in the rotational state of a system, just as force induces change in the translational state of a system. Since the net force on a particle may be connected to the acceleration of the particle by the equation

F

=
m

a

{\displaystyle \mathbf{F}=m\mathbf{a}}

, one may hope to construct a similar relation connecting the net rotational force on a particle to the rotational acceleration of the particle. That may be done as follows:

First, substituting

F

=
m

a

{\displaystyle \mathbf{F}=m\mathbf{a}}

 into the above equation for rotational force, one gets

# τ
m
(

r

×

a

# )
m

r
2

(

r

×

a

r
2

)

{\displaystyle \boldsymbol{\tau}=m(\mathbf r\times \mathbf a)=mr^2(\frac{\mathbf r\times \mathbf a}{r^2})}

.
But from the previous section, it was derived that

# α

r

×

a

r
2

−

2
r

d
r

d
t

ω

{\displaystyle \boldsymbol{\alpha}=\frac{\mathbf r\times \mathbf a}{r^2}-\frac{2}{r}\frac{dr}{dt}\boldsymbol{\omega}}

,
where

α

{\displaystyle \boldsymbol{\alpha}}

 is the orbital rotational acceleration of the particle and

ω

{\displaystyle \boldsymbol{\omega}}

 is the orbital rotational velocity of the particle. Therefore, it follows that

τ

=
m

r
2

(
α

2
r

d
r

d
t

ω
)

=
m

r
2

α
2
m
r

d
r

d
t

ω
.

{\displaystyle \begin{align}
\boldsymbol{\tau} &= mr^2(\boldsymbol{\alpha}+\frac{2}{r}\frac{dr}{dt}\boldsymbol{\omega})\\
\\
&=mr^2\boldsymbol{\alpha}+2mr\frac{dr}{dt}\boldsymbol{\omega}.
\end{align}}

In the special case where the distance

r

{\displaystyle r}

 of the particle from the origin does not change with time, the second term in the above equation vanishes and the above equation simplifies to

# τ
m

r
2

# α
I
α

{\displaystyle \boldsymbol{\tau} = mr^2\boldsymbol{\alpha} = I \boldsymbol{\alpha}}

,
which can be interpreted as a "rotational analogue" to

F

=
m

a

{\displaystyle \mathbf{F}=m\mathbf{a}}

, where the quantity

m

r
2

{\displaystyle mr^2}

 (known as the rotational mass of the particle) plays the role of the mass

m

{\displaystyle m}

. However, unlike

F

=
m

a

{\displaystyle \mathbf{F}=m\mathbf{a}}

, this equation is *not* applicable to an arbitrary trajectory. In conclusion, the general relation between rotational force and rotational acceleration is necessarily more complicated than that for force and linear acceleration.

## See also

* Rotational force
* Rotational momentum
* Angular speed
* Rotational velocity