Introduction:
A conical pendulum is a type of pendulum where the bob (sphere) moves in a horizontal circle instead of swinging back and forth. It is formed by suspending a heavy small-sized sphere from a string of length 'l' and allowing it to rotate uniformly in a horizontal circle at an angle 'θ' with the vertical. The time period of the conical pendulum is the time taken for the sphere to complete one full revolution.

Deriving the Time Period:
To determine the time period of the conical pendulum, we need to analyze the forces acting on the sphere.

1. Tension in the String:
The tension in the string provides the necessary centripetal force to keep the sphere in circular motion. It can be resolved into two components:
- Horizontal Component (T_x): This component balances the centrifugal force acting outward and is given by T_x = T * sin(θ), where T is the tension in the string.
- Vertical Component (T_y): This component counteracts the weight of the sphere and is given by T_y = T * cos(θ), where T is the tension in the string and cos(θ) is the component of gravity acting in the vertical direction.

2. Centripetal Force:
The centripetal force required to keep the sphere in circular motion is given by F_c = m * v^2 / r, where m is the mass of the sphere, v is the linear velocity of the sphere, and r is the radius of the circular path.

3. Equating Forces:
Equating the horizontal component of the tension with the centripetal force, we have:
T * sin(θ) = m * v^2 / r

4. Relation between Velocity and Time Period:
The linear velocity of the sphere can be expressed in terms of the time period (T) and the radius of the circular path (r) using the formula v = 2πr / T.

5. Substituting and Simplifying:
By substituting the value of v in the equation from step 3, we get:
T * sin(θ) = m * (2πr / T)^2 / r
T * sin(θ) = 4π^2 * m * r / T
T^2 = (4π^2 * m * r^2) / (T * sin(θ))

6. Time Period of the Conical Pendulum:
Rearranging the equation, we can solve for T:
T^3 = (4π^2 * m * r^2) / sin(θ)
T = (4π^2 * m * r^2 / sin(θ))^(1/3)

Conclusion:
The time period of the conical pendulum is given by T = (4π^2 * m * r^2 / sin(θ))^(1/3), where m is the mass of the sphere, r is the radius of the circular path, and θ is the angle the string makes with the vertical.