# Show that the wheel has constant retardation using calculus

1. Jul 12, 2012

### MathsRetard09

1. The problem statement, all variables and given/known data

I am required to use calculus to show that a flywheel has constant retardation

Prior to this I have found the initial angular velocity which is 60rad/s and the angular velocity after 2s which is 58.667rad/s

2. Relevant equations

pi=60t-(t^2)/3

dpi/dt = 60 - 2t/3

3. The attempt at a solution

I didn't use calculus at the time because i found it easier to do this owever didn't get an accurate answer:

If t = 4s then:

dpi/dt = 60-2(4)/3

If t = 6s then:

dpi/dt = 60-2(6)/3

At this point i discovered a pattern so immediatly doubled the 6s to 12s to see if i got another even answer, thefore if t = 12s then:

dpi/dt = 60(12)/3

if t = 18s then:

dpi/dt = 60 - 2(18)/3

So i know every 6s the flywheel retards by 4rad/s

However this isn't the correct approach and isn't using calculus and im having trouble trying to understand my notes when i did do this months ago.

Please help me out, it's simple stuff but I just can't for the life of me remember how to do this.

2. Jul 14, 2012

### BruceW

I'm guessing 'pi' means the angular position of some arbitrary point on the flywheel. And so dpi/dt is the angular speed of the flywheel.

The problem is not too difficult. The main thing is to define what is meant by the 'retardation of the flywheel'. Think of some physical concepts which the retardation might correspond to. To be honest, the 'retardation of the flywheel' could mean one of several things, so the question is not worded very well. But from the wording of the question, you can probably guess what the 'retardation of the flywheel' is supposed to mean, physically.

3. Jul 14, 2012

### HallsofIvy

Staff Emeritus
It would help a lot if you defined your terms better- Bruce W assumes that is your angle (measured from where?)- pi is not a good choice. And you don't say what "retardation" means. I think you mean that the bottom of the wheel is moving backwards relative to the axle but that is obviously true.

4. Jul 17, 2012

### MathsRetard09

Figured it out guys.

I wasn't using the calculus

Retardation is basically decceleration - therefore it's acceleration so the value would be negative.

Sorry not pi, sigma - thats my mistake appolagees

Basically Angular Acceleration (alpha) = d2sigma/dt^2

Which basically means that to find the Angular Acceleration I would have to find the second derivative of the original equation I was given:

sigma = 60t - (t^2)/3

which gives me -2/3

So I have my answer but how do I explain that from my workings out that I understand that the wheel has a constant retardation?

Or in other words, the angular acceleration is -2/3 m/s^2 - how do I show that this is going to be constant over time?

Do I need to look at velocity?

$-\frac{2}{3} rad.s^{-2}$ (note the units!) is a constant because it doesn't depend on time. Expressions that are dependent on time would have a 't' somewhere in them, but this does not. So, it's a retardation because of the minus sign, and it's constant because it's independent of t. Your job is done.