Question

Derive an expression for time period of conical pendulum

Answer 1

Derive an expression for time period of conical pendulum

Consider a conical pendulum with a bob of mass m, length l, at an angle θ with the vertical, going round with a uniform velocity v and radius r, as shown in the figure.

The forces acting on the mass are the tension on the string T′, and the force of gravity mg.

The vertical component of the tension balances the force of gravity and the horizontal component provides the centripetal force to keep the bob in uniform circular motion.

вe нappy............^_^

Answer 2

"Time taken by the bob of a conical pendulum to complete one horizontal circle" is called the period of a conical pendulum.

The period of a conical pendulum:

• The vertical component of the tension balances the force of gravity and the horizontal component provides.
• The centripetal force to keep the bob in a uniform circular motion.

Derive an expression for the period of a conical pendulum:

        $v=\sqrt{rg tan}$∅

∴ ω ≈ $\sqrt{} \frac{g tan}{r}$   [∵ v= rω] ..  (1)

       In Δ SOP, $tan$∅$\frac{r}{h}$

from Equation (1),

     ω = $\sqrt\frac{gr}{rh}$

∴      ω = $\sqrt \frac{g}{h}$

ii. If the period of conical pendulum is T

then,

       ω = $\frac{2\pi }{T}$

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

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

iii.   Also, In Δ SOP,

     $cos$∅ $= \frac{h}{l}$

     ∴$h = cos$∅  

where l = length of the string

            ∅ = angle of inclination

Substituting j in equation (2),

  $T = 2\pi\sqrt{x} \frac{l cos}{g}$

This is a required expression for some time of a conical pendulum.