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A particle moves in a plane under the action of a force which is always perpendicular to the particle's velocity and depends on a distance to a certain point on the plane as 1//r^eta, where eta is a constant. At what value of eta will the motion of the particle along the circle be steady? |
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Answer» Solution :This is not central FORCE problem unless the path is a circle about the said point. Rather here `F_t` (TANGENTIAL force) VANISHES. Thus equation of motion becomes, `v_t=v_0=const ant` and, `(mv_0^2)/(r)=(A)/(r^2)` for `r=r_0` We can consider the latter equation as the equilibrium under two forces. When the motion is perturbed, we write `r=r_0+x` and the net force acting on the particle is, `-(A)/((r_0+x)^n)+(mv_0^2)/(r_0+x)=(-A)/(r_0^n)(1-(nx)/(r_0))+(mv_0^2)/(r_0)(1-(x)/(r_0))=-(mv_0^2)/(r_0^2)(1-n)x` This is OPPOSITE to the displacement x, if `n lt 1`. (`(mv_0^2)/(r)` is an outward DIRECTED centrifugul force while `(-A)/(r^n)` is the inward directed external force). |
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