Answer to Question #97426 in Mechanics | Relativity for Mimo

Question #97426
Sketch the orbit described by r = kθ, k is a constant and find the force law for a
central force field that govern this motion.
1
Expert's answer
2019-10-28T11:37:50-0400

We can use the formula for trajectory in the central fields


φ+φ0=Lr2dr2m(EU)L2r2\varphi + {\varphi _0} = \int {\frac{{\frac{L}{{{r^2}}}dr}}{{\sqrt {2m(E - U) - \frac{{{L^2}}}{{{r^2}}}} }}}

Here EE - the full energy, LL - angular momentum, U(r)U(r) - potential energy. The derivation of this equations you may find, for example, in the book "Mechanics: Volume 1 (Course of Theoretical Physics), Landau L.D., Lifshitz E.M.", in paraghaphs 13-14-15.

Now using r=kφr = k\varphi and letting φ0=0{\varphi _0} = 0 we get


Lr2dr2m(EU)L2r2=k1\frac{{\frac{L}{{{r^2}}}dr}}{{\sqrt {2m(E - U) - \frac{{{L^2}}}{{{r^2}}}} }} = {k^{ - 1}}

Thus


k2L2r4=2m(EU)L2r2{k^2}\frac{{{L^2}}}{{{r^4}}} = 2m(E - U) - \frac{{{L^2}}}{{{r^2}}}

And

U(r)=E12m(k2L2r4+L2r2)U(r) = E - \frac{1}{{2m}}({k^2}\frac{{{L^2}}}{{{r^4}}} + \frac{{{L^2}}}{{{r^2}}})

Then the force can be found as F(r)=U\vec F(r) = - \nabla U and


F(r)=(L2mr3+2k2L2mr5)er\vec F(r) = - (\frac{{{L^2}}}{m}{r^{ - 3}} + \frac{{2{k^2}{L^2}}}{m}{r^{ - 5}}){{\vec e}_r}

The trajectory r=kφr = k\varphi is called Archimedean spiral (at the pircture below we use k=1k=1)





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