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=∫2m(E−U)−r2L2r2Ldr
Here E - the full energy, L - angular momentum, 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=kφ and letting φ0=0 we get
2m(E−U)−r2L2r2Ldr=k−1
Thus
k2r4L2=2m(E−U)−r2L2
And
U(r)=E−2m1(k2r4L2+r2L2)
Then the force can be found as F(r)=−∇U and
F(r)=−(mL2r−3+m2k2L2r−5)er
The trajectory r=kφ is called Archimedean spiral (at the pircture below we use k=1)
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