A test particle is moving in a circular orbit in the gravitational field produced by a mass density ρ(r)=(K/r2). Identify the correct relation between the radius R of the particle's orbit and its period T :

Question

A test particle is moving in a circular orbit in the gravitational field produced by a mass density ρ(r)=(K/r2). Identify the correct relation between the radius R of the particle's orbit and its period T : (A) T / R 2 is a constant (B) TR is a constant (C) T2 / R3 is a constant (D) T R is a constant

Tardigrade Admin · Accepted Answer

m =∫ limits0 R ρ 4 π r 2 dr ⇒ m =4 π KR v propto √4 π K ⇒ ( T / R )=(2 π/√4 π K )

Q. A test particle is moving in a circular orbit in the gravitational field produced by a mass density $ρ (r) = \frac{K}{r ^{2}}$ . Identify the correct relation between the radius $R$ of the particle's orbit and its period $T$ :

$T / R^{2}$ is a constant

TR is a constant

$T^{2} / R^{3}$ is a constant

T\R is a constant

$m = 0 \int R ρ 4 π r^{2} d r$
$\Rightarrow m = 4 π K R$
$v \propto 4 π K$
$\Rightarrow \frac{T}{R} = \frac{2 π}{4 π K}$

Q. A test particle is moving in a circular orbit in the gravitational field produced by a mass density ρ(r)=r2K​. Identify the correct relation between the radius R of the particle's orbit and its period T :

T/R2 is a constant