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/R2 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 = ∫R0 ρ4π r2dr m = 4π KR v propto √4π K (T/R) = (2π/√4π 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

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

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

TR is a constant

T2/R3 is a constant

T/R is a constant