What is the relationship between torque and angular momentum?

In summary, Euler's equations state that the angular momentum (and other vector quantities) is conserved between two frames of reference that are rotating with each other.
  • #1
CraigH
222
1
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:

Angular momentum = L
Torque = T
Moment of Inertia = I
Angular Acceleration = α
Velocity = V
Angular Velocity = ω
Radians = θ
Force = F
Time = t

Angular velocity and acceleration
ω=Δ θ/Δt
α= Δ θ/((Δt)^2)

Newton's 2nd law angular form T=I α

Definition of angular momentum L=I ω

I=T/α
I=L/ ω
T/α = L/ ω

T/(Δ θ/((Δt)^2)) = L/(Δ θ/Δt)

T(Δ θ/Δt) /(Δ θ/((Δt)^2)) =L
T/(1/ Δt)=L
T Δt =L

L=T Δt

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

Thanks
 
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  • #2
From a physics textbook:

TΔt = IcΔω, where

T = Torque
t = time
Ic = mass moment of inertia about the center of mass\
ω = angular velocity
 
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  • #3
Thanks :) I just needed someone to confirm I have been using a correct equation.
 
  • #5
CraigH said:
Thanks :) I just needed someone to confirm I have been using a correct equation.
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)##.
 
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  • #6
D H said:
##\vec\tau_\text{ext} = \dot{\vec L} = I\dot {\vec\omega}+ \vec\omega\times(I\vec\omega)##.

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
CraigH said:
Where does this new definition come from?
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
[tex]\left( \frac {d\vec q} {dt} \right)_I = \left( \frac {d\vec q} {dt} \right)_R + \vec \omega \times \vec q[/tex]
Plug in the angular momentum and you get Euler's equations.
 

1. What is torque?

Torque is a measure of the force that causes an object to rotate around an axis or pivot point. It is a combination of the force applied to an object and the distance from the axis of rotation.

2. What is angular momentum?

Angular momentum is a measure of the amount of rotational motion of an object. It is the product of an object's moment of inertia and its angular velocity.

3. How are torque and angular momentum related?

Torque and angular momentum are related through Newton's Second Law of Motion, which states that the net torque applied to an object is equal to the rate of change of its angular momentum.

4. What is the formula for calculating torque?

The formula for torque is τ = r x F, where τ is torque, r is the distance from the axis of rotation, and F is the force applied. It is a vector quantity, meaning it has both magnitude and direction.

5. How does changing torque affect angular momentum?

Changing the torque applied to an object will cause a change in its angular momentum. The direction of the torque will determine the direction of the change in angular momentum, while the magnitude of the torque will determine the magnitude of the change in angular momentum.

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