Why is 2theta(radians) being used to find the angular acceleration.

[gibson101]

I understand how the angular velocity was gotten, and why the amount of revolutions it took to travel 115 m was just 115/circumference of wheel. But why would radians need to be configured to find the angular acceleration? I thought angular acceleration was Δω/τ? And why is 2(thetasymbol) being substituted for time T? And why is the 2 there?

 

[gibson101]

i see that the equation for find angular acceleration was derived from ω^2=ω°^2+2αt. But why can't the formula ω=ω°+αt be used. I noticed that when i used the first formula i ended up with -305/((2∏)(53.8)) = -.9 which is correct but when i used the second formula I got
-.073 which is incorrect. And why does t(time) equal ((2∏)(number of revolutions))?

 

[Redbelly98]

i see that the equation for find angular acceleration was derived from ω^2=ω°^2+2αt.

Not quite, the correct formula is ω²=ω°²+2αθ, where θ is the angle through which the wheel rotates.

But why can't the formula ω=ω°+αt be used. I noticed that when i used the first formula i ended up with -305/((2∏)(53.8)) = -.9 which is correct but when i used the second formula I got
-.073 which is incorrect. And why does t(time) equal ((2∏)(number of revolutions))?

It doesn't. The angle θ through which the wheel rotates is 2π(number of revolutions).

 

[gibson101]

I SEE. BUT i still don't see how θ replaces time. because angular acceleration is Δω/Δt, not Δω/Δθ.

 

[vela]

It's not replacing the time. There are two different formulas:
$$\alpha = \frac{\omega-\omega_0}{t}$$and$$\alpha = \frac{\omega^2-\omega_0^2}{2\theta}$$which allow you to solve for the angular acceleration when the acceleration is constant. The solution used the second equation to find the angular acceleration.

 

[Redbelly98]

I SEE. BUT i still don't see how θ replaces time. because angular acceleration is Δω/Δt, not Δω/Δθ.

It doesn't replace time. θ and time are different concepts, I never said they were the same.

You're correct that angular acceleration is Δω/Δt. But I never said or implied that it is Δω/Δθ, what makes you think I did?

But why can't the formula ω=ω°+αt be used.

If you're trying to find α, but don't know t yet, then that formula is not useful -- even though it is true.

 

[SammyS]

It's not replacing the time. There are two different formulas:
$$\alpha = \frac{\omega-\omega_0}{t}$$and$$\alpha = \frac{\omega^2-\omega_0^2}{2\theta}$$which allow you to solve for the angular acceleration when the acceleration is constant. The solution used the second equation to find the angular acceleration.

In fact, the second equation can be written as:

$$\alpha = \frac{(\omega-\omega_0)(\omega+\omega_0)}{2\theta}$$

$$\phantom{x}\ \ \ = \frac{(\omega-\omega_0)}{\left(\frac{2\theta}{\omega+\omega_0} \right)}$$

So it;s more like t is replaced by θ/((ω+ω₀)/2), which comes from, average (angular) velocity = the total (angular) displacement / elapsed time .