Under the influence of a central potential, a particle moves in a specific orbit. In this case, the particle's orbit is given by r = kθ^4, where k is a constant and r is the distance from the origin. We need to explain how the angle θ varies with time.

1. Central Potential:
- A central potential refers to a force that acts on a particle towards a fixed center, such as gravitational or electrostatic forces.
- In this case, the central potential is unspecified, but we can assume it follows an inverse square law, such as the gravitational force.

2. Polar Coordinates:
- To describe the motion of a particle in a central potential, polar coordinates are used.
- In polar coordinates, the position of the particle is described by the radial distance (r) from the origin and the angle (θ) it makes with a reference direction.

3. Orbit Equation:
- The given orbit equation r = kθ^4 describes the relationship between the radial distance r and the angle θ.
- The exponent of 4 indicates that the radial distance varies with the fourth power of the angle.

4. Deriving θ as a Function of Time:
- To understand how θ varies with time, we need to relate θ to the time variable.
- We can use the concept of angular velocity (ω) to establish this relationship.
- Angular velocity is defined as the rate of change of angle with respect to time, i.e., ω = dθ/dt.

5. Differentiating the Orbit Equation:
- By differentiating the orbit equation r = kθ^4 with respect to time, we can relate ω and θ.
- Taking the derivative of both sides, we get dr/dt = d(kθ^4)/dt.
- The left side represents the radial velocity (v_r), and the right side represents the rate of change of r with respect to time.

6. Angular Velocity and Differentiation:
- Since the radial distance r is a function of θ and time, we need to use the chain rule to differentiate the right side of the equation.
- Applying the chain rule, we have dr/dt = 4kθ^3 * dθ/dt.
- Comparing this with the left side, we can equate the two expressions: v_r = 4kθ^3 * dθ/dt.

7. Equating Angular Velocities:
- Now, we equate the expression for angular velocity derived from the differentiation with the expression v_r = r * dθ/dt.
- Equating the two expressions, we have v_r = 4kθ^3 * dθ/dt = r * dθ/dt.

8. Simplifying and Solving for dθ/dt:
- By canceling out dθ/dt from both sides of the equation, we can solve for it.
- Simplifying the equation, we get 4kθ^3 = r.
- Rearranging, we have dθ/dt = r / (4kθ^3).

9. Conclusion:
- The equation dθ/dt = r / (4kθ^3) represents how the angle θ varies with time in the given orbit.
- This equation shows that the rate of change of θ with time depends on the radial distance r and the constant k.
- By solving this equation, we can find the exact relationship between