Step-by-Step Derivation:

1. Analyze the forces on a particle: Consider a particle of mass m at a radial distance r from the center of the spherical cloud. It is moving in a circular orbit with a certain speed v. For this circular motion to be stable, the gravitational force $F_g$ acting on the particle must provide the necessary centripetal force $F_c$. The gravitational force is due to the total mass M(r) enclosed within the sphere of radius r. According to Newton's shell theorem, the force is: $F_g = \frac{G M(r) m}{r^2}$ The centripetal force required for a circular orbit of radius r at speed v is: $F_c = \frac{m v^2}{r}$

2. Apply the force balance equation: Equating the gravitational force and the centripetal force: $F_g = F_c$ $\frac{G M(r) m}{r^2} = \frac{m v^2}{r}$

3. Incorporate the kinetic energy information: The problem states that all particles in the cloud have the same kinetic energy K. $K = \frac{1}{2} m v^2$ From this, we can express $mv^2$ in terms of the constant K: $m v^2 = 2K$

4. Solve for the enclosed mass M(r): Substitute $mv^2 = 2K$ into the force balance equation: $\frac{G M(r) m}{r^2} = \frac{2K}{r}$ Now, we can solve for M(r): $G M(r) m = 2Kr$ $M(r) = \frac{2Kr}{Gm}$

5. Relate enclosed mass to mass density: The mass M(r) enclosed within a radius r is the integral of the mass density $\rho(r)$ over the volume of the sphere. A small mass element dM in a spherical shell of radius x and thickness dx is $dM = \rho(x) dV = \rho(x) (4\pi x^2 dx)$. So, $M(r) = \int_0^r \rho(x) 4\pi x^2 dx$ To find $\rho(r)$, we can differentiate M(r) with respect to r: $\frac{dM(r)}{dr} = \rho(r) 4\pi r^2$ This gives us an expression for $\rho(r)$: $\rho(r) = \frac{1}{4\pi r^2} \frac{dM(r)}{dr}$

6. Calculate the mass density $\rho(r)$: Differentiate the expression for M(r) from Step 4: $\frac{dM(r)}{dr} = \frac{d}{dr} \left( \frac{2Kr}{Gm} \right) = \frac{2K}{Gm}$ Now substitute this derivative into the equation for $\rho(r)$: $\rho(r) = \frac{1}{4\pi r^2} \left( \frac{2K}{Gm} \right) = \frac{2K}{4\pi r^2 Gm} = \frac{K}{2\pi r^2 Gm}$

7. Calculate the particle number density n(r): The particle number density n(r) is defined as the mass density $\rho(r)$ divided by the mass of a single particle m. $n(r) = \frac{\rho(r)}{m} = \frac{1}{m} \left( \frac{K}{2\pi r^2 Gm} \right)$ $n(r) = \frac{K}{2\pi r^2 G m^2}$

8. Compare with options: The derived expression is $n(r) = \frac{K}{2\pi r^2 m^2 G}$. This matches option D.

Conclusion:

The particle number density n(r) is given by $\frac{K}{2\pi {r^2}{m^2}G}$. Therefore, option D is the correct answer.