Conical pendulum

3 forces acting on a conical pendulum

• Gravity(=mg): The force the Earth pulls.
• Tension: The force applied to both ends of the thread. Tension is always directed to the center from both ends.
• Centripetal force(Fc): The force that maintains circular motion. The centripetal force always toward the center of rotation.

$$F_{c} = mr \omega ^{2} = mr {( \frac{2\pi }{T})}^2$$

ω: Angular velocity of rotational motion (rad/s)
T: period of rotational motion (seconds)

Find the period of a conical pendulum

Consider a conical pendulum with a line length ‘L’ and a rotation radius ‘r.’
Centripetal force(Fc) is the result of gravity and tension. Also, the centripetal force is perpendicular to gravity. From this, the following equation is calculated.

$$F_{c} = mg \cdot tan \theta$$

Substituting the definition of centripetal force comes the following equation.

$$mr {( \frac{ 2 \pi }{T})}^2 = mg \cdot tan \theta$$

$${( \frac{ 2 \pi }{T})}^2 = \frac {g }{r} \cdot tan \theta$$

Since it is $tan \theta = \frac{r}{h}$, we can make it a bit simpler.

$${( \frac{ 2 \pi }{T})}^2 = \frac {g}{h}$$

Solving the above equation for the period T, It becomes as follows.

$$T = 2 \pi \sqrt{\frac {h}{g}}$$