The Physics Philes, lesson 40: What’s Your Angle?

In which radians are converted, angular velocity and acceleration are calculated, and I figure out what a flywheel is.

Alrighty, folks. Last week we laid the foundation for understanding angular velocity and acceleration. Now it’s time to apply that knowledge with some practice problems. Whoop whoop!

I guess we should just jump right in.

We’re testing a flywheel. (What’s a flywheel?) The diameter of the flywheel is 0.36 m and the angular position θ is:

Screen shot 2013-03-17 at 10.17.24 AM

What can we figure out from this information? Let’s try to find the angle θ at 2.0 s and 5.0 s. All we have to do in this case is plug either 2.0 s or 5.0 s into the given equation. We’ll call the angle θ at 2.0 s θ1 and the angle θ at 5.0 s θ2.

Screen shot 2013-03-17 at 10.32.54 AM

Screen shot 2013-03-17 at 10.33.06 AM

Hm. OK. What the heck does 16 rad and 250 rad mean? Sounds like a lot of radians. Maybe? Maybe if we convert these values to degrees it will be easier to comprehend.

Screen shot 2013-03-17 at 11.01.10 AM

Screen shot 2013-03-17 at 11.01.15 AM

Whoa. OK, that puts radians into perspective. OK, now let’s see if we figure out how far a particle on the rim of the flywheel moves in the time interval from 2.0 s to 5.0 s. Remember that the formula for this is s = rθ, or arc length equals the radius of the circle times the angle. So all we need to figure this out is the radius of the flywheel and the difference between angle θ at 2.0 s and 5.0.

We know the diameter of the flywheel, so we also know the radius. Just divide by two! So r = 0.18 and Δθ = 234 rad. All we need to do is plug the values into the equation:

s = (0.18 m)(234 rad) = 42 m

Remember, a radian is a pure number with no unit. I just put rad in there to make it clear that that’s what we’re dealing with.

Now let’s find the average angular velocity. The formula for average angular velocity is ω = Δθ/Δt. We know the two values of θ and the two values of t, so we can easily calculate the average angular velocity:

ω = Δθ/Δt = (250 rad – 16 rad) / (5.0 s – 2.0 s) = 78 rad/s

Score! Look at all the stuff we did with only a little bit of information! Science is awesome. Let’s do another problem. This time, let’s try to calculate angular acceleration.

Let’s say the instantaneous angular velocity of the flywheel can be given by this equation:

Screen shot 2013-03-17 at 11.51.30 AM

Let’s find the angular acceleration between 2.0 s and 5.0 s.

Average angular acceleration is defined as the difference in angular velocity divided by the difference in time. So first we need to figure out the angular velocity. But we already know how to do that!

Screen shot 2013-03-17 at 11.56.57 AM

Screen shot 2013-03-17 at 11.57.06 AM

Now that we know the angular velocity, we can find the average angular acceleration.

Screen shot 2013-03-17 at 12.00.12 PM

Woo hoo! We solved the average angular acceleration!

We did it. We calculated angular velocity and acceleration, and we did a little conversion, too. You should be very proud of yourselves. Next week we’ll talk about rotation with constant acceleration. It’ll be awesome sauce.

Featured image credit: Wikimedia Commons

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Mindy is an attorney and Managing Editor of Teen Skepchick. She hates the law and loves stars. You can follow her on Twitter and on Google+.

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