Question

Show that the angular momentum is conserved under central force

Answer 1

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A problem I am trying to work out is as follows:

A particle moves in a force field given byF⃗ =ϕ(r)r⃗ F→=ϕ(r)r→. Prove that the angular momentum of the particle about the origin is constant.

I set it up as follows:

F⃗ =md2r⃗ dt2F→=md2r→dt2

v⃗ =∫F⃗ m dt=∫ϕ(r)r⃗ m dtv→=∫F→m dt=∫ϕ(r)r→m dt

which is equal to :

ϕ(r)tr⃗ m+cϕ(r)tr→m+c

(I am not sure what I am doing at this point. Is my integrated expression correct?)

Assuming it is, we get:

Angular Momentum L=m(r⃗ ×v⃗ )=r⃗ ×(ϕ(r)tr⃗ +c)L=m(r→×v→)=r→×(ϕ(r)tr→+c)

Now I don't know what to do with the constant term, but I do know that

r⃗ ×kr⃗ =0r→×kr→=0

However, the problem states that we have to prove the result is a constant, so I think I'm wrong. Specific places where someone could help me out are:

(1) Is my integration correct? If not, how does one integrate a force (given in terms of position vector notation) w.r.t. time?

(2) What happens to the constant? Cross-product of a vector and a scalar doesn't make any sense
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