Angular Acceleration Of A Wheel

[Warmacblu]

Homework Statement

A wheel rotating with a constant angular acceleration turns through 22 revolutions during a 9 s time interval. Its angular velocity at the end of this interval is 15 rad/s.

What is the angular acceleration of the wheel? Note that the initial angular velocity is not zero.

Homework Equations

w = omega
a = alpha

w_f = w_i + at
theta = w_it + 1/2at²

The Attempt at a Solution

w_f = 15
w_i = ?
a = ?
theta = 2pi * 22 = 128.23
t = 9

So I have two unknowns and two equations but I can't seem to solve for the two equations. Both equations have two unknowns in them which is throwing me off. Any help is appreciated.

Thanks

 

[rl.bhat]

The first equation can be written as
wf*t = wi*t + a*t^2and subtract it from the second equation.wi*t get canceled out.Then solve for a.

 

[Warmacblu]

How does this look?

w_ft = w_it + at²
theta = w_it + 1/2at²

w_ft - theta = at² - 1/2at²

w_ft - theta / t² = a - 1/2at²

w_ft - theta / t⁴ = a - 1/2a

2w_ft - theta / t⁴ = a

Thanks for the help.

 

[rl.bhat]

w_ft - theta = at² - 1/2at²

w_ft - theta / t² = a - 1/2at²
This step is wrong. It should be
w_ft - theta = 1/2*at²

 

[Warmacblu]

w_ft - theta = at² - 1/2at²

w_ft - theta / t² = a - 1/2at²
This step is wrong. It should be
w_ft - theta = 1/2*at²

How come the at² can just be dropped?

Following your correction:

w_ft - theta = 1/2*at²

w_ft - theta / t² = 1/2a

2w_ft - theta / t² = a

 

[rl.bhat]

How come the at² can just be dropped?

Following your correction:

w_ft - theta = 1/2*at²

w_ft - theta / t² = 1/2a

2w_ft - theta / t² = a

It should be
2[w_ft - theta] / t² = a[

 

[Warmacblu]

It should be
2[w_ft - theta] / t² = a[

Here is what I got:

2((15*9)-138.23) / 9² = -.0798

Does that mean that the disc has a negative acceleration?

Also, I still do not understand how you just dropped the at²:

wf_t - theta = at² - 1/2at²

wf_t - theta / t² = a - 1/2at²
This step is wrong. It should be
wf_t - theta = 1/2*at²

 

[rl.bhat]

Also, I still do not understand how you just dropped the at^2
I am not dropping at^2. It is a simple algebra.
wft-θ = at^2 - 1/2*at^2
= 2*1/2*at^2 - 1/2at^2...= ? [Just like (2x - x) = ?]

 

[Warmacblu]

Also, I still do not understand how you just dropped the at^2
I am not dropping at^2. It is a simple algebra.
wft-θ = at^2 - 1/2*at^2
= 2*1/2*at^2 - 1/2at^2...= ? [Just like (2x - x) = ?]

Ah, okay. I see it now.

I am unsure if my answer is in rad/sec² which the question specifies for. I multiplied 15 rad/s times 9 sec - 138.23 rad all divided by 9 sec. The terms seem correct but the negative is still throwing me off.

 

[rl.bhat]

See whether the angular velocity increasing or decreasing. If it is decreasing then the acceleration is negative.

 

[Warmacblu]

22 rev / 9 sec = 2.44 rev/sec = 15.36 rad / sec. At the end of the time interval it is at 15 rad/s, so I guess the acceleration is negative.