Pretty certain that this concern has already been reply in this site, yet I cannot uncover it. Anyway, here"s the question: What is the difference in between angular momentum and moment the inertia?


Angular momentum is the "moment that momentum", meaning it gives us one idea the how far is the linear momentum vector used at. Torques show off the moment arm of a force, and also angular momentum requires the minute arm of momentum.

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Particle Mechanics

Take a solitary particle moving in a directly line (in the lack of external forces). It has mass $m_i$, it is located at vector $oldsymbolr_i$ through velocity vector $oldsymbolv_i$. This sets united state up because that the complying with definitions

Linear Momentum that particle, $$ oldsymbolp_i = m_i oldsymbolv_i $$Angular Momentum about the origin, $$ oldsymbolL_i = oldsymbolr_i imes oldsymbolp_i $$

The over is sufficient to recuperate the place of the route line, at the very least the suggest on the route of the fragment closest come the origin.

$$ oldsymbolr_ m path = frac oldsymbolp_i imes oldsymbolL_i oldsymbolp_i $$

You can easily prove this if you show that $oldsymbolL_i = oldsymbolr_ m path imes oldsymbolp$, which you execute with vector triple product identity $oldsymbol a imes( oldsymbol b imes oldsymbol c) = oldsymbol b (oldsymbol a cdot oldsymbol c) - oldsymbol c ( oldsymbol a cdot oldsymbol b)$.

So in summary, Angular momentum describes the (permendicular) distance where linear momentum acts. The conservation of angular inert law method that not only direct momentum as a vector is conserved, but likewise the ar of this vector (or the line in space the vector action through) is conserved.

Rigid body Mechanics

When you extend the above to multiple particles clumped together as a rigid human body the principle of moment of inertia arises. An initial off, Charles"s to organize state the for all the distances to it is in maintained, each particle can only relocate with a mix of translation and also rotation (with vector $oldsymbolomega$) about a common axis. Typically the facility of mass is supplied as a reference point, and so the motion of each bit in the body is limited to $oldsymbolv_i = oldsymbolv_ m com + oldsymbolomega imes oldsymbolr_i$.

Commonly the motion is decomposed come the translate in of the center of mass, and also the rotation about the center of mass. This returns the complying with relationships

Linear Momentum of strictly body, $$oldsymbolp = sum limits_i m_i oldsymbolv_i = m oldsymbolv_ m com$$

Angular Momentum about center that mass, $$oldsymbolL_ m com = sum limits_i oldsymbolr_i imes m_i oldsymbolv_i = sum limits_i m_i oldsymbolr_i imes ( oldsymbolomega imes oldsymbolr_i ) $$

Mass moment of inertia

In bespeak to recognize angular momentum of a rigid body rotating about the center of massive better, the is typical to separate the geometry components from the activity parts

$$ oldsymbolL_ m com = underbrace mathrmI_ m com _ extgeometry ;;underbrace oldsymbolomega_ extmotion $$ where

$$ mathrmI_ m com = sum_i (-m_i <oldsymbolr_i imes><oldsymbolr_i imes>) = sum_i m_i left| matrix y^2+z^2 & - x y & - x z \ - x y & x^2+z^2 & -y z \ -x z & -y z & x^2+y^2 ight| $$

This is the mass minute of inertia tensor. The is the rotational equivalent to mass, due to the fact that $oldsymbolp = m oldsymbolv_ m com$ and $oldsymbolL_ m com = mathrmI_ m com oldsymbolomega$ have actually a comparable form.

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So, mass minute of inertia defines how much away from the rotation axis is the mass distributed at. That conveys the geometry info of angular momentum. For this reason if a known mass moment of inertia about an axis $I$ is defined by $I = m r ^2$, it means the geometry that the trouble is similar to that of a massive ring through radius $r$.