Fan of this fan

Let the angular velocity be $\omega$

Given that velocity becomes half after 36 rotation $i.e.$ after angular displacement of $36 \times 2 π$

So

$\frac{\omega ^2 }{4} = \omega ^ 2 - 2 \alpha 36 \times 2π$ $\alpha \text{ is angular deceleration}$ $\Rightarrow \alpha = \frac{\omega ^ 2}{192π}$ Now let the number of more rotations be $n$ $0= \frac{\omega ^ 2 }{4} - 2 \frac{\omega ^ 2}{192π} x 2π$ $\Rightarrow x=12$

Here's a more conceptual solution:

Let $\omega$ be the angular velocity. The total number of rotations in an interval of time is $x = C \times \int_{t_1}^{t_2}\omega\ dt$ where $C$ is some constant depending on your units. On a graph of $\omega\text{ vs. } t,$ the integral is just the area of a right triangle. Also, the angular acceleration is constant, so it takes the same amount of time to slow down from full speed to half speed as it does to slow down from half speed to rest. See it now? The integral on the left (from full to half speed) is just three times the integral on the right (from half speed to zero). If $t$ is the time from full to half speed, then $\int_0^t\omega\ dt = 3\int_t^{2t}\omega\ dt$ $x_1 = 3x_2$ Now substitute $x_1 = 36$ to get $x_2 = \boxed{12}$

Since the fan rotates 36 times before it's angular speed is down to a half, we know that the angular position can be expressed as

$θ = 72π$ rad

Thus the angular velocity at that moment can be expressed as

$ω2 = 72π/t$ where t is the time it takes to rotate $72π$ $rad$

Then from this, we know that the initial angular velocity is twice that value given above so

$ω1 = 144π/t$

Let $u$ = ω1 then we have:

$\frac{1}{2}$ $u^2$ - $u^2$ = $2α72π$

$\frac{-3}{4}$ $u^2$ = $144πα$ rearranging for α we get $α = -u^2/192π$

Thus $α = -108π/t^2$

This time let $ω3 = 0$ as the final angular velocity when the fan stops, then we would get the equation to be:

$0 - u^2 = 2αθ$ where θ represents the total number of rotations until the fan stops permanently

Substituting all the values and rearranging for θ we get,

$θ =$ $\frac{u^2t^2}{216π}$

Expand this expression by using the known value for $u$ . This would result in,

$θ = 96π$ But this is the total amount. We need the remaining amount which we get to be,

$96π - 72π = 24π rad$

Converting this to rotations we get $24π / 2π = 12$ more rotations.

Therefore the fan will stop completely after 12 more rotations.

here is a simple method

every one knows the equations

v2 - u2=2as (v2 & u2 are squares of v & u)

use it in angular form and get the answer

first find 'a' in terms of 'u' ( 2a=u2/48)

then substitute 2a and find 's'(no of rotations) for 'v'=0&'u'=u/2