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How I determine the center of mass of a semicircle using the definition of center of mass? I only know solve this using the Pappus theorem. Consider that the semicircle is centered on the origin and a homogeneous mass distribution.

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closed as too localized by user1504, Waffle's Crazy Peanut, Manishearth Apr 10 '13 at 16:18

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Hi Example Mo. Welcome to Physics.SE. This site deals with conceptual Physics Q&A. We don't encourage homework questions that doesn't involve any sort of work done by the author (which is you) and asks other users to solve the problem. If you think you could clarify your question, add what you've done along with your question. We're ready to help you. If you aren't clear, Please have a look at our homework policy for more info. After improving the post, flag it for moderator attention. –  Waffle's Crazy Peanut Apr 10 '13 at 15:01
thank you for the comment, i will improve the post –  Example Mo Apr 10 '13 at 15:08
Do you mean a semi-circle or a semi-disk? –  Qmechanic Apr 10 '13 at 15:27
In the question i need of a semi-circle, but now i am trying to use the Bru answer to figure out how to solve the semi-disk. –  Example Mo Apr 10 '13 at 15:33
Welcome to Physics! Please see our homework policy. We expect homework problems to have some effort put into them, and deal with conceptual issues. If you edit your question to explain (1) What you have tried, (2) the concept you have trouble with, and (3) your level of understanding, I'll be happy to reopen this. (Flag this message for ♦ attention with a custom message, or reply to me in the comments with @Manishearth to notify me) –  Manishearth Apr 10 '13 at 16:17

2 Answers 2

You can determine the center of mass of an arbitrary object by integrating over its volume:

$$\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV.$$

In your case, where mass is distributed homogeneously, $\rho$ is a constant.

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In the case of a one dimensional object, the center of mass $\vec r_{\text{CM}}$, if given by $$ M \vec r_{\text{CM}} = \int_{C} \vec r \text{d}m $$ where $M$ is the total mass (it is given by the linear density multiplied by the length of the semi-circle), $C$ denotes the semi-circle and $\vec r$ is the vector locating a point on $C$. You should first choose appropriate coordinates for you problem, and then express the quantities appearing in the integral in these coordinates. You'll see that the integral is then very easy to compute.

Good luck !

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Hi Bru. This isn't a site for solving homework problems. I think your answer is helping the author who hasn't actually tried anything..! –  Waffle's Crazy Peanut Apr 10 '13 at 15:05
I agree. Sorry for this. I was just imaginating myself as a first-year student, completely lost in front of such problems, and I though I would have been happy to find such a detailed answer. But I understand that this is not the goal of this forum. –  Bru Apr 10 '13 at 15:14
This is not a very big mistake and so, there's no need to apologize. Hope you can spend a couple of minutes in our homework policy ;-) –  Waffle's Crazy Peanut Apr 10 '13 at 15:21
Thanks Bru, i understand what you do!! If i need to calculate the center of mass of a filled semicircle, i need to use double integral? I wanted to apologize for not having explained what I had done in the exercise, but i promise my next questions will be written correctly. –  Example Mo Apr 10 '13 at 15:22
@ExampleMo : I think that given all the details of my answer and the answer of Frederic Brünner above you have enough informations to answer this question by yourself - if not be specific about your problem. –  Bru Apr 10 '13 at 15:29

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