Wednesday, 23 March 2016

homework and exercises - Proving that vecvtimessumidfracdmidtvecri=t(vecvtimessumivecFi) when no external torque


There is this idea of relativity in Classical Mechanics:



The laws of mechanics valid in an inertial frame must also be valid in any frame moving uniformly with respect to it.



I was just trying to apply these to the case of the law of conservation of momentum and the law of conservation of angular momentum.


Let there be an inertial frame S and another frame S' moving with velocity v w.r.t to S with:


ri=rivt



vi=viv


For momentum conservation: In frame S', putting ddtipi=0 and substituting ddtipi=0 of frame S in it:


ddtipi=ddtipiddtimiv=0vddtimi


If this has to be 0, then imi=0


Now, on to angular momentum. In frame S:


ddtiLi=ddti(ri×mivi)=0


Am trying to prove the law in frame S' from the law in S:


ddtiLi=ddtiLiddti(ri×miv)ddti(vt×mivi)


=0ddti(ri×miv)ddti(vt×mivi)


=imi(vi×v)idmidt(ri×v)+imi(vi×v)imi(vt×ai)idmidt(vt×vi)



=idmidt(ri×v)imi(vt×ai)idmidt(vt×vi)


=v×idmidtrivt×iFi


But this is what I wanted to prove to be 0. I stil have to prove the following:



For a system of particles at ri with mass mi, which have forces Fi acting on them such that iri×Fi=0, given idmidt=0; how do I prove:


v×idmidtri=vt×iFi


for any arbitrary v and for all time t.




Answer



Using your notation of



ri=rivtvi=viv


and with the assumption that ˙v=0 (uniform motion of frame S') form the linear and angular momentum expressions on the S frame.


p=imiviL=i(ri×mivi)


Now look at linear and angular momentum in the S' frame and relate them to the ones from S.


p=imivi=imi(viv)=p(imi)v=pmvL=i(ri×mivi)=i(rivt)×mi(viv)=i(ri×mivi)i(ri×miv)vt×(imivi)+vt×(imi)v=L+v×i(miri)+i(mivi)×vt


To show that these quantities are conserved, take the derivative (assuming that dpdt=0 and that dLdt=0)


ddtp=ddtpmddtv=0ddtL=ddtL+ddt[v×i(miri)]+ddt[i(mivi)×vt]=v×i(middtri)+ddtp×vt+i(mivi)×v=v×i(mivi)+p×v=v×p+p×v=0


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