A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is used to describe its motion, the radius r of the n^th orbit and the particle's speed v in the orbit depend on n as

The correct option is:
$r\propto n^{\frac{2}{3}},v\propto n^{\frac{1}{3}}$

Step-by-Step Explanation:

Step 1: Understand the forces acting on the particle
The particle of mass m is moving in a circular orbit of radius r around the origin. A constant force F pulls the particle towards the origin, which acts as the centripetal force holding the particle in its circular path.

Therefore, we can write the equation of motion as:
$\frac{m{v}^2}{r}=F$

Since both the mass m of the particle and the force F are constant, we can establish a proportionality relation between the velocity v and the radius r:
$v^2\propto r$
Taking the square root on both sides gives:
$v\propto r^{\frac{1}{2}}$ ---- (Equation 1)

Step 2: Apply Bohr's angular momentum quantization condition
According to the Bohr model, the angular momentum L of the particle in the n^th orbit is quantized and given by:
$mvr=\frac{nh}{2\pi }$

Since m, h, and 2π are constants, the product of velocity and radius is directly proportional to the principal quantum number n:
$vr\propto n$ ---- (Equation 2)

Step 3: Determine the dependence of radius r on n
Substitute the proportionality of v from Equation 1 into Equation 2:
$r^{\frac{1}{2}}\cdot r\propto n$
Simplify the powers of r:
$r^{\frac{3}{2}}\propto n$
Raising both sides to the power of 2/3, we get the dependence of the orbit radius r on n:
$r\propto n^{\frac{2}{3}}$ ---- (Equation 3)

Step 4: Determine the dependence of velocity v on n
Now, substitute the relation for r from Equation 3 back into Equation 1:
$v\propto {\left(n^{\frac{2}{3}}\right)}^{\frac{1}{2}}$
Simplifying the exponents:
$v\propto n^{\frac{2}{3}\cdot \frac{1}{2}}$
$v\propto n^{\frac{1}{3}}$

Thus, we find that:
$r\propto n^{\frac{2}{3}}\text{and}v\propto n^{\frac{1}{3}}$