I know that angular momentum can be expressed in terms of moment of inertia tensor as follows, L → = I tensor w → My question is how to get this definition from tensor definition of it ?

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I know that angular momentum can be expressed in terms of moment of inertia tensor as follows,            L      →        =      I          tensor                  w      →      My question is how to get this definition from tensor definition of it ?

suprakarmay5 · Accepted Answer

The moment of inertia describes an objects tendency to remain rotating. We can think of it on similar terms to mass for momentum, where mass gives an object a greater tendency to continue its motion.For a point mass rotating about some axis at distance r, we have that   I  =  m      r    2  Now if we extend this concept to be the moment of inertia for a system of three point masses, at distances       r    1    ,      r    2    ,      r    3   from the rotation axis, we sum the moments for each mass as:  I  =      ∑    k              m      k              r      k      2        =      m    1        r    1    2    +      m    2        r    2    2    +      m    3        r    3    2  The above is the definition you are seeing in your text. Now we can apply this concept in a more useful form, where we are summing up individual point masses within a continuous mass distribution. We write:  I  =  ∫      r    2    d  mwhere our limits of integration are across the whole continuous mass.As you stated, we can write             L      →       in tensor form:            L      →        =            I      ~                  ω      →                 →               [                                        L            x                                                            L            y                                                            L            z                                ]    =      [                                        I                          x              x                                                            I                          x              y                                                            I                          x              z                                                                        I                          y              x                                                            I                          y              y                                                            I                          y              z                                                                        I                          z              x                                                            I                          z              y                                                            I                          z              z                                            ]        [                                        ω            x                                                            ω            y                                                            ω            z                                ]  Where             I      ~       represents the moment of inertia tensor matrix above. We can write each I from their definitions (      I          x      x       and       I          x      y       as examples):      I          x      x        =      ∑    k    (      y    k    2    +      z    k    2    )      m    k    =  ∫  (      y    2    +      z    2    )  d  m      I          x      y        =      I          y      x        =  −      ∑    k        x    k        y    k        m    k    =  −  ∫      x    y         d    m

What's exactly is moment of inertia?

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