# What is the relationship between torque and angular momentum?

1. Aug 13, 2013

### CraigH

I'm looking for an equation similar to the change in forward momentum equation:

Δmv=f*Δt

But for angular momentum.

I think it will be (change in angular momentum) = (torque) * (change in time)

Here is how I derived it:

Is this all correct? I cannot find this equation anywhere on the internet but it seems right.

Thanks

2. Aug 13, 2013

### mark.watson

From a physics text book:

TΔt = IcΔω, where

T = Torque
t = time
Ic = mass moment of inertia about the center of mass\
ω = angular velocity

3. Aug 13, 2013

### CraigH

Thanks :) I just needed someone to confirm I have been using a correct equation.

4. Aug 14, 2013

### A.T.

5. Aug 14, 2013

### D H

Staff Emeritus
The equation posted, $\vec\tau_\text{ext} = \dot{\vec L} = I\dot {\vec\omega}$, is fine for a first year student. You are apparently a third year student now, so you shouldn't be using that equation anymore. It's a "lie-to-children." You should be using Euler's equations instead, $\vec\tau_\text{ext} = \dot{\vec L} = I\dot {\vec\omega}+ \vec\omega\times(I\vec\omega)$.

6. Aug 15, 2013

### CraigH

torque = rate of change of angular momentum, this is the same as $\vec\tau Δt = Δ\vec L$ isn't it?

So the bit I am getting wrong is my definition of angular momentum.

My definition comes from the angular momentum of a particle http://hyperphysics.phy-astr.gsu.edu/hbase/amom.html#amp
$\vec L = \vec r X \vec p$
this makes sense, if r is in the same direction as p (linear momentum) then the angular momentum will be 0, if it is orthogonal to p then it will be maximum.

Where does this new definition come from?

7. Aug 15, 2013

### D H

Staff Emeritus
The transport theorem. http://en.wikipedia.org/wiki/Rotating_reference_frame#Time_derivatives_in_the_two_frames.

The angular momentum of a rigid body is the product of the body's moment of inertia tensor about the center of mass and the body's angular velocity: $\vec L = I\vec\omega$. Trick question: What frame is it expressed in?

All of the standard formulae for the inertia tensor (e.g., http://en.wikipedia.org/wiki/List_of_moment_of_inertia_tensors) are expressed in rotating frame coordinates (a frame rotating with the rigid body). The moment of inertia tensor in inertial coordinates is a time-varying beast; it's constant in this rotating frame. The answer to my trick question is rotating frame coordinates. Working in inertial coordinates here would be insane.

On the other hand, the expression $\vec\tau_{\text{ext}}=\frac{d\vec L}{dt}$ is about what's happening in the inertial frame. How to relate the time derivative of the angular momentum in the inertial frame to that in the rotating frame? That's what the transport theorem does. For any vector quantity $\vec q$ the transport theorem says
$$\left( \frac {d\vec q} {dt} \right)_I = \left( \frac {d\vec q} {dt} \right)_R + \vec \omega \times \vec q$$
Plug in the angular momentum and you get Euler's equations.