Is this Wikipremed answer wrong?

[Geekchick921]

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If so, I'm going to e-mail him, I just want to make sure before I do.

It's the last question in this module video. http://wikipremed.com/course_videos.php?syl=02&video_code=010105_01

Bicycle brakes are being checked at a repair shop. The brakes deliver an 8N friction force to the 1.0 m diameter wheel rim. If the wheel is rotating at 10 rad/s and its moment of inertia is 2.0 kg m^2, how long does it take for the wheel to come to rest?

A. 2.5 seconds
b. 5 seconds
c. 10 seconds
d. 20 seconds

He said B. I think it's A. I pause the video after the problem comes up to see if I get it right before he starts explaining it, and we were on the same page until he calculated the angular acceleration. I think he goofed in converting it from meters to radians.

T = F d = (8N)(0.5m) = 4N m
T = I α Solve for α.
(4N m)/(2.0 kg m^2) = 2 m/s^2

But since the radians are 0.5 m, it's 4 rad/s^2, right? This is where we deviated. He just said it was 2 rad/s^2, and the answer would be 5 sec. But if the angular acceleration is 4 rad/s^2, it would be 2.5 sec. He's been awesome so far and this is the first error I noticed. I started watching the next video to see if he addressed it and so far he hasn't.

So... who's right?

 

[Pre-Med Village]

Hi Geekchick921. This is John Wetzel from the WikiPremed MCAT course. I noticed the post so I thought I'd drop in. Thank you for the nice thing you said about the course. It means a lot to me to hear that.

The particular Rotation problem is also in the physics cards on the site: http://wikipremed.com/01physicscards.php?card=246

I think what happened is that you have slightly misremembered the formula relating torque, moment of inertia and angular acceleration.

Here is what you wrote:

T = F d = (8N)(0.5m) = 4N m
T = I α Solve for α.
(4N m)/(2.0 kg m^2) = 2 m/s^2

You found a translational acceleration here, a, when the answer should be angular acceleration, α. So this shouldn't be 2 m/s^2 but 2 rad/s^2. To put it in plain language, a net torque produces an angular acceleration. So if the angular velocity is decreasing according to the angular acceleration by two radians per second per second it will take five seconds to deplete the initial angular velocity of 10 rad/s to zero.

I think this is right. I hope this is helpful.

 

[Geekchick921]

Hi John, I've seen you post here before. Thank you for responding!

I think I'm just confused by the units, since radians aren't part of the torque or moment of inertia units. Since I put the solution I came to in m/s^2, I thought it needed to be converted to radians, which would effectively double it.

 

[Pre-Med Village]

I comment sometimes around here, if a commenter brings up WikiPremed, though I am very conservative about commenting at studentdoctor.net because I don't want to harm the forum by promoting things.

Regarding your question, let me say that if we were in seminar I would be very happy about the reason for the confusion, because dimensional analysis is a really valuable conceptual shorthand in physics to have. Possessing the habit of keeping the units of the physical quantities in mind in physics will do you a great deal of good. If everyone really thought about what it means for a volt to be a joule per coulomb, for example, a lot of electricity and magnetism would immediately be more coherent.

Basically, the crux here is that a radian is not a unit like a meter or a second. It's a pure number. We get taught to say that a radian is the angle subtended by an arc equal in length to the radius, but it may help you to think about the radian measurement as a ratio. It's the ratio between the arc length and the radius. What do you get if you divide a length by a length? You get a dimensionless ratio. Does that make sense? That's why you don't see the radian in either the units for torque or moment of inertia. It's not that kind of unit.

This is not the kind of thing that will come up often. It's not a big deal as mistakes go. Thinking about the units is really good. This is the first time I'v seen it lead to problems.

 

[Geekchick921]

That makes sense, thank you! I am overthinking it. I really appreciate you taking the time to clarify. 🙂

 

[Pre-Med Village]

That's it. Please stay in touch and let me know how the course is going. All the best.