Rotation of a Rigid Object around a fixed axis

[K.S]

As a result of friction, the angular speed of a wheel changes with time according to

dθ/dt = ω_o*e(-σt)

where ω_o and σ are constants. The angular speed changes from 3.50 rad/s at t=0 to 2.00 rad/s at t=9.30s.

(a) Use the information to determine σ and ω_o. Then determine
(b) the magnitude of the angular acceleration at t=3.00s,
(c) the number of revolutions the wheel makes in the first 2.50s, and
(d) the number of revolutions it makes before coming to rest.

Now, I can get part (a) to (c) - my answers are:
(a) ω_o = 3.50; σ=0.0602
(b) -0.176 rad/s^2
(c) 1.29 revs

However, I have in part (d) the expression
ω_o*e(-σt) = 0 ,

but I find there is no way I can get an answer from this expression, since ln0 is undefined.

Can anyone enlighten me please? Thanks in advance!

Source: Physics for Scientists and Engineers with modern physics, 8th edition, Serway and Jewett, page 318

 

[Filip Larsen]

You are correct that the angular speed as defined in this problem never reaches zero, but the question relates to the angular position θ. Perhaps if you look at the expression for θ it will be more obvious what to do.

 

[Doc Al]

However, I have in part (d) the expression
ω_o*e(-σt) = 0 ,

What are you trying to do with this expression?

 

[K.S]

Filip Larsen, I did as you have suggested, and I obtained

θ = ω_o/σ (1 - e(-σt)) , when I integrated ω from t=0 to t=t

Now I have the variable t which i had attempted to find from the expression ω_o*e(-σt).

Doc Al, I had wanted to find the time t when the wheel comes to rest - or as I had interpreted as when angular speed becomes zero.

Inferring from your reactions, I think there is something gravely wrong with the assumption that angular speed becomes zero when the wheel comes to a rest? Sorry all, it does seem my foundation is a little shaky. :x

 

[Doc Al]

Doc Al, I had wanted to find the time t when the wheel comes to rest - or as I had interpreted as when angular speed becomes zero.

As Filip points out, it never does come to rest. (It's an exponentially decreasing function.) But that won't stop you from finding the angle traveled through. So what should be the time when it comes to rest? (That will give you your limits of integration for finding the angle.)

Inferring from your reactions, I think there is something gravely wrong with the assumption that angular speed becomes zero when the wheel comes to a rest?

No, that's what coming to rest means. Nothing wrong with that!

 

[K.S]

Oh god, thanks Doc Al! You're a lifesaver. Okay I got it, and I did so by finding

lim->inf ω_o/σ (1 - e(-σt)) = ω_o/σ

Thanks Filip Larsen for the help too!

 

[Filip Larsen]

I obtained θ = ω_o/σ (1 - e(-σt)) , when I integrated ω from t=0 to t=t

Now I have the variable t which i had attempted to find from the expression ω_o*e(-σt).

In your expression for θ above, what happens when t goes to infinity?

Inferring from your reactions, I think there is something gravely wrong with the assumption that angular speed becomes zero when the wheel comes to a rest? Sorry all, it does seem my foundation is a little shaky. :x

There is nothing wrong as such. You can calculate the position angle of the wheel when it comes to rest, even if "when" here means after infinitely long time. You should probably think of this problem more of an exercise in mathematical concepts (e.g. limits) than you should consider it an example of a physically accurate model.