Help with a certain part of Center of Mass problems...(concept)?

Question

Hi, I'm learning Center of Mass in physics right now, and although I got the points made in the lecture, I can't seem to do the hw. I don't need the answer, but I want to know how certain things given in the problem play a role in the answer. Here's the problem for reference:

A square uniform raft, 18 m by 18 m, of mass 6200kg, is used as a ferryboat. If three cars, each of mass 1200kg, occupy thte NE, SE and SW corners, determine the CM of the loaded Ferryboat.

Now, my teacher said put the origin in someplace easy, so I chose the center of the ferryboat as (0,0). I went through the calculations, and got: Xcm = 1.1 and Ycm = -1.1

However, when I put the origin at the South East Corner, I got Xcm = -5.69 and Ycm = 5.69. (Am I allowed to put the origin at this place?)

Is there something wrong here? Also, I would also like to know if the mass of the ferryboat matters here...I think it does. Please help!!!

Answer 1

The mass of the ferryboat indeed matters in this problem. Imagine if they weren't cars, but rather feathers? The center of mass would be virtually unchanged in position from the center of the ferryboat.

You're allowed to put the origin whereever you wish. Remember, the laws of physics should be the same no matter what (inertial) frame of reference you choose.

Some hints:
The cars in the NE and SW corners balance each other out, and have the same center of mass as the boat. So you can treat that boat as carrying a single car in the SE corner, with the boat having an additional 2400kg of mass. But you probably already knew that. Incidentally, I agree with your Xcm = 1.1 m, and Ycm = -1.1 m. Make sure you state your dimensions when citing answers.

Answer 2

The coordinates of a point change when you change the origin. For example, I am at (0,0,0) with respect to my place on the couch I'm on, but at (10,0,0) with respect to an origin ten feet to my left. It's very important to specify your coordinate system before you calculate. For example, (0,0) at the center of the boat with +x toward the East and +y toward the North.

Here's the important part: coordinates are just names for a point--not the point itself. Different coordinate systems will give different coordinate answers, but these apparently different answers are just DIFFERENT NAMES (coordinates) for the SAME POINT. You can check two different coordinate system answers against each other by taking the coordinates from one, and translating them into the other system. This is called a "coordinate transformation" and I'm sure your teacher will talk about it.

The mass of the ferryboat will not matter in the numerator if you put the origin exactly at the center of mass of the ferryboat itself. Then everything is measured with respect to that point, and since the ferryboat's mass is exactly at (0,0) it exerts no moment about that point. But it still goes in the denominator. So this would in fact make the problem a little simpler, and indeed the CM of a good heavy ferryboat should be pretty much right near this origin if the boat is not to tip too much (see the feather argument above).

When you put the origin at the SE corner, you make the moment of the SE car at zero, but then the ferryboat will matter in the numerator as well. This is because it exerts a large moment about the relatively off-center point.

Answer 3

Yes, the mass of the ferry boat is critical.

If the origin is at 0,0
- you have a 6200 kg mass at 0,0
- you have 1200 kg masses at (9,9) and (-9,-9) These average to (0,0), so add another 2400 kg to (0,0)
- you have 1200 kg mass at (9,-9)

so, 8600 at zero and 1200 at (9,-9). This makes the center (1.125,-1.125)

If you make the center at the southeast corner, you end up with the same stuff but a different reference. you have 8600 kg at -9,9 and 1200 at 0,0. The answer is now (-7.875,7.875). This is the same point as above just called a different name due to where you put the origin.