Conical pendulum

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Simple pendulum

A pendulum is a body suspended from a fixed support that swings freely back and forth under the influence of gravity. Pendulums are used to regulate devices such as clocks.

Conical pendulums are similar to simple pendulums; however, instead of rocking back and forth, a conical pendulum moves at a constant speed in a circle with the string tracing out a cone.

From the point of view of an external observer, there are two external forces acting on a conical pendulum bob, which should be considered with the pendulum bob's resistance to change of motion.

External forces:

the tension in the string which is exerted along the line of the string
the downward action of gravity on the mass of the bob

Tension in the string can be resolved vertically and horizontally:

horizontal - Tsinθ (acting towards the centre of the circle)
vertical - Tcosθ

Since the system is balanced, the forces are equal in magnitude to their opposites. In other words:

the vertical component of the tension in the string acting vertically upwards is equal and opposite to the weight of the bob acting vertically downwards
the horizontal component of the tension in the string (the centripetal force acting inwards) is equal and opposite to the resistance of the bob to changes in motion (acceleration) relevant for the circular motion (the centrifugal force acting outwards).

$T Sin \theta\ = \left ( \frac {mv^2}{r} \right )$ and $T Cos \theta\ = mg$

Furthermore, one can find the height that the pendulum traces in terms of the period.

$T = \left(\frac {mg} {h}\right)$ Tension in the pendulum is equal to the force of gravity on the bob divided by the vertical distance of the bob to the origin of the pendulum.

$\frac{mg} {Cos\theta} = \frac {mv^2} {rSin \theta}$ Since the tension is equal in magnitude but opposite in direction, the two equations can be set equal, m will cancel out.

$v = \left ( \frac {2\pi r}{t} \right )$ This equation represents the period of the pendulum, where t is the time the pendulum takes to complete one revolution in minutes, and v is velocity in revolutions per minute.

$\frac{g}{Cos\theta} = \frac {v^2}{rSin\theta} = \left ( \frac {2\pi r}{T} \right )\ \left ( \frac {1}{Sin\theta} \right )^2\$ This is a simplification of setting all expressions equal to each other. It is a way to parameterize the equations.

$\frac {g}{Cos\theta} = \frac{v^2}{rSin\theta} = \left (\frac {4\pi^2}{T^2} \right )\left (\frac{r}{Sin\theta}\right ) = \frac{4\pi^2}{T^2}\left (l\right)$

$g \frac{1}{Cos\theta} = g \left(\frac{l}{h}\right )$

$\left(g\frac{l}{h}\right ) = \left (\frac {4\pi^2}{T^2} \right )$

$h = \frac {T^2}{4\pi^2}g$

These steps refer back to the equations set next to the diagram. They are used to simplify the equation so the height of the pendulum can be expressed in terms of its period.