Question

When a ceiling fan is switched off. It's angular velocity falls to half while it makes 36rotations .how many more rotations will it make before coming to rest?

Answer 1

hey there is answer !!!
==================
I hope you help !!!
==============
mark me brain list friends !!!
=========================


now the answer is started !!!
=======================



Let its initial angular velocity be ω , when it becomes half its original value i.e. ( ω / 2 ) 
it makes 36 rotations that means 36 * 2 π  rad 
Hence , from ( ω / 2 ) 2 = ( ω ) 2 - 2 ​α  θ 
we get an expression as 
 2 ​α  θ = ( 3 / 4 ) ω 2   Plug  θ = 36 * 2 π       ........... ( 1 ) 
Now , when ​ω = 0 it comes to rest 
So, again from ( ω / 2 )2 -  2 ​α  θ = 0 
we get ( 1 / 4 ) ω 2 = 2 ​α  θ          
          ​θ = ( 12  * 2 π)  or 12 rotations     [  from ( 1 )   ] 
Hence , proved that it will rotate 12 times more before it comes to rest .

Answer 2

Question :

When a ceiling fan is switched off. It's angular velocity falls to half while it makes 36rotations .how many more rotations will it make before coming to rest?

Solution :

case 1 :

Let the initial angular velocity of fan be ω.

Now when switch is off , final angular velocity ω'= ω/2

$\sf\:\theta=2\pi\:\times36$

Here α is constant ( angluar acceleration)

Now use Equation of motion for circular motion .

$\sf\:\omega{}^{2}_{f}=\omega{}^{2}_{i}+2\alpha\:\theta$

$\sf\:\implies\:\dfrac{\omega{}^{2}}{4}-\omega{}^{2}=2\alpha\:\theta$

$\sf\:\implies\:\alpha=\dfrac{-\omega{}^{2}}{192\pi\:}...(1)$

Case 2:

Now initial angular velocity= ω/2

and final angular velocity= 0

Use Equation of motion in circular motion

$\sf\:\omega{}^{2}_{f}=\omega{}^{2}_{i}+2\alpha\:\theta$

$\sf\:\implies\:0-\dfrac{\omega{}^{2}}{4}=2\alpha\:\theta$

Now put the value of α from equation (1)

$\sf\:\implies\:-\dfrac{\omega{}^{2}}{4}=2\times\:\dfrac{-\omega{}^{2}}{192\pi\:}\times\:\theta$

Therefore ,

$\bf\:\theta=24\pi$

So, number of rotations made by the fan before coming to rest

$\sf\:=\dfrac{24\pi}{2\pi}=12\:rotations$