Name: Rectilinear Motion and Its Example
Uploaded: 2019-07-19
Duration: 11 min 6 s
Description: What is the similarity between the motion of people marching and the motion of a ball rolling down an inclined plane? Let's get an answer in this video.

Question

What is a conical pendulum? Show that its time period is given by

2 π \sqrt{\frac{lcosθ}{g}}

, where

l

is the length of the string,

θ

is the angle that the string makes with the vertical and

g

is the acceleration due to gravity.

Accepted Answer

In a simple pendulum a mass $m$ is suspended by a string of length $l$ and moves along an arc of a vertical circle. If the mass instead revolves in a horizontal circle and the string which makes a constant angle with the vertical describes a cone whose vertex is the fixed point $θ$ , then mass-string system is called a conical pendulum.

As shown in the figure, the forces acting on the bob of mass $m$ of the conical pendulum are:

(i) Gravitational force, mg, acting vertically downwards

(ii) Force due to tension T acting along the string directed towards the support.

hence, we resolve T in two components

from the fig,

$Tcosθ = mg$

The resultant force on the bob is then $Tsinθ$ which is radial or centripetal or directed towards center O'

$T \sin θ = \frac{m v^{2}}{r} = m r ω^{2}$

$\tan θ = \frac{m v^{2} / r}{m g} = \frac{v^{2}}{r . g}$

we know $v = \frac{2 πr}{T}$

$tanθ = \frac{4 π^{2} r^{2}}{T^{2} . r . g} T = 2 π \sqrt{\frac{r}{gtanθ}}$

we know $r = lsinθ$

$T = 2 π \sqrt{\frac{l . sinθ}{gtanθ}} = 2 π \sqrt{\frac{l . cosθ}{g}} h = l . cosθ \Rightarrow T = 2 π \sqrt{\frac{h}{g}}$

What is a conical pendulum? Show that its time period is given by $2 π \sqrt{\frac{lcosθ}{g}}$ , where $l$ is the length of the string, $θ$ is the angle that the string makes with the vertical and $g$ is the acceleration due to gravity.

Important Questions on Motion in a Plane

Learn and Prepare for any exam you want !

What is a conical pendulum? Show that its time period is given by 2πlcosθg, where l is the length of the string, θ is the angle that the string makes with the vertical and g is the acceleration due to gravity.

Important Questions on Motion in a Plane

Learn and Prepare for any exam you want !

What is a conical pendulum? Show that its time period is given by $2 π \sqrt{\frac{lcosθ}{g}}$ , where $l$ is the length of the string, $θ$ is the angle that the string makes with the vertical and $g$ is the acceleration due to gravity.