# Does angular acceleration not always come with torque?

1. Jan 5, 2010

### carrotstien

I always thought that

torque = I*d²θ/dt²

so that, if there is any d²θ/dt² on an object with a moment of inertia (both with respect to the same point)..then there must be a torque applied.

However, I've found a case where this isn't true. So, i'm assuming there is more to it then that simple formula.

I found, on wiki
"When the moment of inertia is constant, one can also relate the torque on an object and its angular acceleration in a similar equation:

τ = I α

"

So perhaps, when the moment of inertia isn't constant, then that formula has some stuff appended to it. (what?)

The situation i speak of:
A particle is traveling in a straight line not through the origin at a constant velocity.
It's (r) and (θ) are both functions of time. If you work out the time derivatives, you get a non-zero θ''. However, there are no forces, and no torques on the particle. (like a car just cruising past you).
Also, the origin isn't moving or accelerating, so that there doesn't seem to be some relativity issue (i am not talking about special or general)

Can anyone explain / tell me the full relation between torque and θ''...since
torque = I*θ'' seems to be a simplified case.

2. Jan 5, 2010

### Nabeshin

In analog with newton's second law for translation, the full formula would be something like:
$$\vec{\tau}=\frac{d\vec{L}}{dt}$$
$$\vec{L} \equiv I \vec{\omega}}$$

So only when moment of inertia is constant does it reduce to the more familiar form.

3. Jan 5, 2010

### Bob S

An automobile supplies torque to its wheels at constant velocity to compensate for air drag. So when there is energy loss, torque is required to maintain a constant angular velocity.

Angular acceleration can occur without torque:

dL/dt = d(Iω)/dt = I dω/dt + ω dI/dt

If I dω/dt = - ω dI/dt , no torque is required for angular acceleration.
Bob S

4. Jan 5, 2010

### rcgldr

If the angular inertia is changed via an internal force, then angular acceleration will occur while the angular inertia is changing because angular momentum is conserved. Note that internal work is peformed during such a change, and that angular kinetic energy will also change.

Last edited: Jan 5, 2010
5. Jan 5, 2010

### carrotstien

yea, thanks, Nabeshin and all...i get it
i just never knew that L was defined in terms of Iw the way you showed it.
Now everything makes sense again.

6. Jan 5, 2010

### carrotstien

yea, thanks, Nabeshin and all...i get it
i just never knew that L was defined in terms of Iw the way you showed it.
Now everything makes sense again.