Explanation:

The given equation represents the polar equation of an orbit, where r is the distance from the origin and θ is the angle. Let's analyze this equation in detail:

1. Definition of a central potential:
A central potential is a force field where the force acting on a particle is always directed towards a fixed point called the center. In this case, the center is the origin (0,0) in a Cartesian coordinate system.

2. Polar coordinates:
In polar coordinates, a point is specified by its distance from the origin (r) and the angle it makes with the positive x-axis (θ). The conversion between polar coordinates and Cartesian coordinates is given by:
x = r cos(θ)
y = r sin(θ)

3. Relationship between r and θ:
The given equation of the orbit is r = kθ, where k is a constant. This equation implies that the distance from the origin (r) is directly proportional to the angle (θ). As the angle increases, the distance from the origin also increases.

4. Time dependence:
To analyze the time dependence of the angle θ, we need to consider the angular velocity (ω) of the particle. The angular velocity is defined as the rate of change of the angle with respect to time:
ω = dθ/dt

5. Deriving the time dependence:
To find the time dependence of θ, we differentiate the given equation with respect to time:
d(r)/dt = d(kθ)/dt
0 = k(dθ/dt)
dθ/dt = 0

From this, we can conclude that the angle θ does not change with time. Therefore, the particle moves at a constant angular velocity and completes its orbit in a fixed amount of time.

6. Interpreting the result:
The result implies that the particle's motion is uniform around the origin. It moves along the orbit with a constant angular velocity, maintaining a fixed distance from the origin.

7. Physical significance:
The equation r = kθ represents a spiral orbit. This type of motion can be observed in various systems, such as the motion of planets around the sun or the motion of electrons around the nucleus in an atom. The constant k determines the shape and size of the spiral orbit.

Overall, the given equation describes the motion of a particle under the influence of a central potential in a spiral orbit, where the angle θ does not vary with time.