Multiple answers to this question? Centripetal acceleration

[MusicTheorist]

Hi, I'm working on a paper about the problems with standardized testing. On a recent test that I took there was one problem that I thought could demonstrate one of the problems with standardized testing, multiple answers to a single question based on interpretation, and I thought I might use it in the paper. However, first I wanted to check to make sure I was right that there were two possible answers.

Homework Statement

We are given a diagram of a cart in centripetal motion. There is no data given (except for the radius=2.0m), but there are arrows indicating the direction of motion in the circle, and a radius is drawn.

The problem says: If the mass of the cart was doubled, the magnitude of the centripetal acceleration would be:

A) doubled
B) unchanged
C) halved
D) quadrupled

Please give your own answer before you look at mine.

Homework Equations

F_c=ma_c
a_c=v²/r

The Attempt at a Solution

I really debated this question for a while. I thought that if we're talking about a_c based on its equation the answer has to be unchanged because mass isn't a factor in determining the acceleration.

However, the first thing that came to my mind was F=ma, and the inverse relationship between mass and acceleration. If the mass doubles the acceleration would have to be halved or else the force would be changed. Although we didn't mention force, I put force into the picture because we were talking about a mass moving with acceleration around a circle, and if we don't know the maximum force possible, increasing the acceleration could send the cart off its path and there would be no centripetal acceleration because it would move off in a line.

I'm curious to know what you would put. I went with halved because I asked my teacher if F_c was remaining constant and she told me it was, but she then later marked the question wrong and said she didn't understand my question (she gave me the point back though when I explained it).

 

[Pengwuino]

I'd like to know what the magnitude of a cart is. Wood? Steel? Are carts complex?

 

[MusicTheorist]

Agh, sorry. Typo. The question should have read:

If the mass of the cart was doubled, the magnitude of the centripetal acceleration of the cart would be:

It's a pretty basic question, I just think its ambiguous.

 

[Pengwuino]

Why would the force have to remain the same? Imagine a planet in a circular orbit. Doubling the mass of the object in the same orbit wouldn't change the velocity or the orbital radius. The acceleration would be the same as well. However, the force would double. It is a different situation, but I hope it illustrates that forces don't have to remain the same for something to remain under the same acceleration.

Your teacher was incorrect, the force would have to change. If the force were to be the same, the acceleration would decrease which would have to result from a change in radii or velocity

 

[Dick]

Agh, sorry. Typo. The question should have read:

If the mass of the cart was doubled, the magnitude of the centripetal acceleration of the cart would be:

It's a pretty basic question, I just think its ambiguous.

It's not ambiguous at all. I think your teacher was wrong if she really said F_c is constant. If only the mass changes, not v or r, F_c would change, wouldn't it? So sure, the acceleration is unchanged.

 

[MusicTheorist]

Yes, I know Fc would change, but here's what I thought:

If there's an object on a string in centripetal motion, and the maximum tension force is x, there's a possibility that doubling the mass without changing the acceleration could make the string snap and then it would just be regular acceleration. I assume that would be the same for a cart in centripetal motion.

So, for whatever reason (I'm not sure why), I just assumed that this was the maximum centripetal force that could be exerted before the cart was sent off in a tangential line.

 

[Dick]

Yes, I know Fc would change, but here's what I thought:

If there's an object on a string in centripetal motion, and the maximum tension force is x, there's a possibility that doubling the mass without changing the acceleration could make the string snap and then it would just be regular acceleration. I assume that would be the same for a cart in centripetal motion.

That's certainly true. Do you think that is what the question is about? If strings start snapping everything changes. What's wrong with answer B?

 

[MusicTheorist]

That's certainly true. Do you think that is what the question is about? If strings start snapping everything changes. What's wrong with answer B?

There's nothing wrong with it, I just thought that the same principle that applies to string tension would apply to anything in centripetal motion, and because of that unless I knew whether the force could change or not I thought I had to ask because otherwise changing the mass could exceed the maximum force. When I asked she said force couldn't change so that's how I got C.

I realize I over thought the question, and it was really extremely basic, but is it invalid to over think it?

 

[Dick]

There's nothing wrong with it, I just thought that the same principle that applies to string tension would apply to anything in centripetal motion, and because of that unless I knew whether the force could change or not I thought I had to ask because otherwise changing the mass could exceed the maximum force. When I asked she said force couldn't change so that's how I got C.

Right. But you already said you thought F_c would change. And I would agree. I already said I don't think your teacher was right if she said F_c wouldn't change. I think we can both agree your teacher might have said a wrong thing, right? Oh, wait a minute. I just went back and reread everything. This isn't about the physics question is it? It's about ways standardized testing can go wrong, isn't it?

 

[MusicTheorist]

Well, I thought Fc COULD change, as long as the force currently being exerted wasn't the maximum, and doubling the mass wouldn't exceed the maximum. The lack of that information is what I think makes the question ambiguous.

I agree that my teacher said something wrong, but that's not really my point in asking this. I was curious to see if anyone thought about the question the same way I did, or if anyone else saw a way the question could be thought of in different ways and get different answers.

 

[Dick]

Well, I thought Fc COULD change, as long as the force currently being exerted wasn't the maximum, and doubling the mass wouldn't exceed the maximum. The lack of that information is what I think makes the question ambiguous.

I agree that my teacher said something wrong, but that's not really my point in asking this. I was curious to see if anyone thought about the question the same way I did, or if anyone else saw a way the question could be thought of in different ways and get different answers.

Yeah, I finally see your point in asking this. Sorry to be dense. It's about pedagogy, not physics. I'm not very comfortable with this example. You say m doubles. But you don't explicitly say what doesn't change. Of course, something else HAS to change. A properly framed question should probably say m changes with v and r fixed. Then there is an unambiguous answer. If you don't specify what doesn't change, then you probably have an infinite number of possible answers that are consistent with m changing, not just two. But it's certainly ambiguous.

 

[MusicTheorist]

Yeah, that was why I asked the question. I'm big into educational reform (I made a website termonline.org and was on the TODAY show talking about it recently), and this is one of the things that I think is wrong with standardized testing: people who understand the material can get things wrong because they think of it in a different way, even if it can be valid.

 

[Dick]

Yeah, that was why I asked the question. I'm big into educational reform (I made a website termonline.org and was on the TODAY show talking about it recently), and this is one of the things that I think is wrong with standardized testing: people who understand the material can get things wrong because they think of it in a different way, even if it can be valid.

Right. But you've said m changes to 2*m. If you also assume F_c doesn't change, then the string won't break. Then r or v must change according to your posted equations. r might change if the string unknots a kink suddenly. v might change if you hit a sticky patch in the track or bounce off an obstacle. Both are pretty artificial. I really wouldn't expect a student who is not totally contrary to bring these possibilities up.There has got to be a better example of a question that could be understood to have two different answers given the students assumptions. If you don't get perversely eccentric, the correct answer is B.

 

[MusicTheorist]

Right. But you've said m changes to 2*m. If you also assume F_c doesn't change, then the string won't break. Then r or v must change according to your posted equations. r might change if the string unknots a kink suddenly. v might change if you hit a sticky patch in the track or bounce off an obstacle. Both are pretty artificial. I really wouldn't expect a student who is not totally contrary to bring these possibilities up.There has got to be a better example of a question that could be understood to have two different answers given the students assumptions. If you don't get oddly eccentric, the correct answer is B.

I'm not sure what you're saying here. If you assume that F_c doesn't change then a_c has to change, but where does the string unknotting a kink or the sticky patch come in? These possibilities aren't why I thought it was ambiguous.

This isn't a string by the way, it's just a cart moving in a circle.

 

[Dick]

I'm not sure what you're saying here. If you assume that F_c doesn't change then a_c has to change, but where does the string unknotting a kink or the sticky patch come in? These possibilities aren't why I thought it was ambiguous.

This isn't a string by the way, it's just a cart moving in a circle.

You said string snapping. I am saying that a reasonable physics student would say the correct answer is B. It's not that bad a question. Saying if the mass doubles the string would break is also outside most peoples reasonable interpretation of the question. TODAY show notwithstanding, this is the sort of arguments I get from students trying to change a C- to a C+. Come on. Find a real ambiguous question. I never claimed there aren't any. This isn't one of them.