In a hypothetical spherical galaxy, the mass density is given by p=k/r If a planet is rotating at R 0 distance from the center of the galaxy. Find the relation between time period T and radius R 0.
Answers
Explanation:
Before arriving at the answer let us first understand the meaning of these terms:-
⭕RADIUS OF CURVATURE:-
✍It is known to us that a spherical mirror has a pole and a point called the "centre of curvature ".
✍This distance between the pole and the centre of curvature is the RADIUS OF CURVATURE.
⭕FOCAL LENGTH:-
✍It is the length between the pole of a spherical mirror (optical centre of lens) and the focal point.
✍It is positive in case of a convex mirror and negative in that of a concave mirror.
▶️Let us now come at the relation between the focal length and centre of curvature.
focal \: length = \frac{1}{2} (radius \: of \: curvtature)focallength=
2
1
(radiusofcurvtature)
⭕One important thing to be noted is that, in case of a plane mirror, the radius of curvature is taken as infinity.
⭕In case of lens ,if you are provided with refractive indices of the medium, you can have the lens maker formula.
\frac{1}{f} = n - 1( \frac{1}{ r_{1} } - \frac{1}{ r_{2} } )
f
1
=n−1(
r
1
1
−
r
2
1
)
where f is the focal length and r1 and r2 are the radius of curvature respectively. ✔✔
In a hypothetical spherical galaxy, the mass density is given by ρ = k/r If a planet is rotating at R₀ distance from the center of the galaxy.
To find : The relation between time period T and radius R₀.
solution : first find mass by this variable density of it.
∫dm = ∫ρ dV
given , ρ = k/r
we know, volume of sphere, V = 4/3 πr³
differentiating w.r.t to r we get,
dV/dr = 4πr² ⇒dV = 4πr² dr
now ∫dm = ∫k/r × 4πr² dr
⇒M = 4πk [r²/2]
⇒M = 4πk × R₀²/2 = 2πkR₀²
at equilibrium,
gravitational force = centripetal force
⇒GMm/R₀² = mw²R₀
⇒GM/R₀³ = w²
⇒(2π/T)² = G(2πkR₀²)/R₀³
⇒4π²/T² = 2πkG/R₀
⇒T = 2π √{R₀/2πkG} = √{2πR₀/kG}
Therefore the relation between time period and R₀ is T = √{2πR₀/kG}