How Many Revolutions Does the Wheel Make in 2.33 Seconds?

[jj8890]

Homework Statement

As a result of friction, the angular speed of a wheel changes with time according to
d θ/d t = ω_0 e^(−σ t) ,
where ω0 and σ are constants.
initial angular speed = 5.45 rad/s
σ =.23913
Determine the number of revolutions the wheel makes after 2.33 s .
Answer in units of rev.

The Attempt at a Solution

I know I have to take the integral which is:
θ = - w_0/(σ *e^(-σ *t))
My numbers are initial velocity= 5.45 rad/sec, sigma= .023913, and time=2.33 seconds. so when I solved I got -13.0589 radians and divided it by 2pi to get revolutions, which gave me -2.07839. I thought the answer would be 2.07839 because revolutions can't be negative but it is not right, should i use the negative or am i doing something wrong?

 

[tiny-tim]

Determine the number of revolutions the wheel makes after 2.33 s .
Answer in units of rev.

{snip} … so when I solved I got -13.0589 radians and divided it by 2pi to get revolutions, which gave me -2.07839. I thought the answer would be 2.07839 because revolutions can't be negative but it is not right, should i use the negative or am i doing something wrong?

Hi jj!

I'm with you on this: the question is badly worded.

The wheel makes 2.07839 revolutions clockwise!

However, that's clearly not the answer they want - if this is a computer thing where you're only allowed to put in a number, then you must put in the number they want - this isn't a constitutional rights case!

But if you're allowed a full answer, write something like "-2.07839 revolutions in the direction of increasing θ (2.07839 revolutions in the direction of decreasing θ)" - with the answer they want first.

 

[Moetasim]

Hello,

Thank you for reaching out. Your approach is correct, but you made a mistake in your calculation. When you integrated the equation, you should have gotten:

θ = -ω0/σ * e^(-σ * t) + C

Where C is the constant of integration. To solve for C, you can use the initial condition given in the problem, where t = 0 and θ = 0. Therefore, we can set up the following equation:

0 = -ω0/σ * e^(-σ * 0) + C

C = ω0/σ

Substituting this value of C back into the equation, we get:

θ = -ω0/σ * (e^(-σ * t) - 1)

Now, plugging in the values given in the problem, we get:

θ = -5.45/.23913 * (e^(-.23913 * 2.33) - 1) = -2.0784 radians

To convert this to revolutions, we divide by 2π, giving us:

θ = -2.0784/2π = -0.3309 revolutions

So the wheel makes approximately 0.3309 revolutions after 2.33 seconds. Since we are measuring angular speed, the negative sign indicates that the wheel is rotating in the opposite direction of the initial angular speed (clockwise instead of counterclockwise). I hope this helps. Let me know if you have any further questions.

Best regards,