The angular equivalent of momentum.

Defined as L=r x p (radius cross product momentum).

### It's the rotational analog of momentum

As force is required to accelerate a mass (change it's velocity) in a linear fashion, torque is required to accelerate something in a rotational fashion (change it's rotational or angular velocity).

Torque is most easily understood by considering how a lever works. The same force, located twice as far from the fulcrum, will create twice the torque, becuase:

Where radius = distance from the axis of rotation

This equation is part of the result from the equation, above, since
p = mv (momentum = mass x velocity) and
Force = dp / dt = m (dv/dt)
where dp/dt = change in momentum, and
dv/dt = change in velocity = acceleration, so

The more that the mass is distributed away from the center of rotation, the harder it is to rotate, by applying a rotational force (torque) at the axis.

This is why it's easier to open a sticky door by pushing further from the hinge, or why you can turn a rusty nut easier it you put a pipe on the end of the wrench.

The rotational analog of mass is the moment of intertia, which is just a measure of how the mass is distributed, with respect to the axis of rotation.

Just as a larger mass takes more force than a smaller mass to get it moving at the same speed, an object with a larger moment of inertia takes more torque to get it rotating at the same angular velocity, as above.

To get an intuitive feel for the difference, consider an umbrella; When it's closed, you can spin it easily - not much torque is required to get it rotating rather quickly (at a high RPM).

Now, open it up, and spin it again - now it's much harder to get it rotating at the same speed. The mass is exactly the same, but it's moment of inertia has increased quite a bit, since its mass is now distributed farther from the axis.

### Angular momentum, like linear momentum, is conserved.

If, as you spin the closed umbrella, it opens up, you'll find that its rotation slows down.

In a similar way, when a spinning ice skater brings her arms in close, her rotation speeds up. Next time you watch a skater, notice how, just before a jump, they'll spread their arms and legs as far out as possible. Then, they dig one skate in, to start rotating, and once airborne, bring their limbs in as close as possible. They are taking advantage of the conservation of angular momentum, to spin as fast as possible, making triple, and even quadruple, rotations possible.

But that's not real easy for untrained folks to do. Fortunately, there is a way you can experiment with the same dynamics.

Next time you are at the playground, find the merry-go-round, and try this:

1. Hop on, grab one of the bars, and lean as far out past the edge as possible
2. Have a friend spin it, or use one leg to spin it - you don't need to spin it very fast, either
3. Once spinning, move toward the center - and hang on!
4. If you try this with multiple people, you'll find it almost impossible to get to the center, as the rotational velocity increases quite a bit, creating a lot of centrifugal force.

You can do similar tricks with two people on a tire-swing, or on roller-skates, too.

Well here is how to prove the Torque-Angular Momentum relation which is the equivalent of F=dp/dt.
We know that
L = r x p
So if we differentiate both sides, we get
dL/dt = dr/dt x p + r x dp/dt
however dr/dt is just v, and as p = mv, therefore v x p = 0. Also using Newton's second law, the second term becomes r x F = T(because that is the definition of torque).
Thus we end up with
dL/dt = T

A few comments. First of all L and T depend not only on the frame of reference you are in but also on the origin you are using . Thus it is very important to specify the origin.
Second, angular momentum shows a neat division just like kinetic energy - we can write
L=r x pcm + Lcm
Where pcm is the momentum of the center of mass and Lcm is the angular momentum about the center of mass in the frame of reference of the center of mass.
Finally, the Torque-Angular momentum relation must be applied in a inertial reference frame. However, there is a very special non-inertial frame in which it also works and that is the center of mass frame(with the origin at the cm). This is a natural consequence of the above splitting of L into two parts.

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