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## MCAT

### Unit 8: Lesson 4

Newton's laws and equilibrium

# Center of mass

Sal gives a conceptual understanding of what center of mass means. Created by Sal Khan.

## Want to join the conversation?

• But how do you really calculate the centre of mass in physics problems? • @ Mat Hildebrand: This method can in fact be generalized to an integral.
In which case it is Xcom=Integral(x*dm)/Integral(dm)
Where dm is the mass of an infinitesimal (meaning really really small) piece of your object, x is the position of that infinitesimal piece, and Xcom is the center of mass. This can then be repeated for the y dimension and the z dimension by simply changing all of the x's to y's or z's respectively.(Note: to actually evaluate the integral, you'll want to convert your dm into terms of dx. Specifically, dm=rho*dx , where rho is the density (mass per length) of your object.
• Is the centre of mass and centre of gravity the same?
Also are they always at the same place?
If not then could you please give and example? •   If the gravitational field is constant, then the centre of mass and centre of gravity will be in the same place. For almost all problems you will ever come across, you can treat them as the same. In reality, they are not quite. For example, because your head is further from the centre of the Earth than your feet are, each kilo of your head experiences less of a gravitational pull than each kilo of your feet. So your centre of gravity will be very slightly lower (more towards the Earth) than your centre of mass is. But the difference is so very, very slight that you can almost always ignore it.
• Wait... how does their center of mass go below the bar? I'm not understanding why it doesn't remain where it originally was? •   If you bend your body, more of your body mass will be in behind you. In the diagram you can see that the guys legs and arms are behind him. So the center of mass is actually outside the persons body. This may seem weird, but think of a donut, its center is hollow, but its center of mass is in its center, even though its empty in the middle
• is it possible the center of mass of an object is outside the body?? •  Yes, it is. As an example, think of a doughnut (or any other type of ring) -- the center of mass is located at the center of the doughnut (where the hole is) and, hence, outside its doughnut body.

I should note, however, that I am assuming we're talking about a regular doughnut that contains a hole in the center (thus excluding all jelly/cream-filled doughnuts that do not contain said holes). This answer is making me hungry... Hope this helps!
• I don't understand how the Center of Mass is coming out of the body when the man is jumping in an arched manner over the pole at . Please help! • centre of mass is a theoretical concept... its major advantage is that it simplifies motion ... like if we are looking for motion of a elephant .. then we can study elephants motion just by seeing motion of its "centre of mass" ...
centre of mass need not always be inside body ... it is generally inside body when body has some regular shape or uniform density ... when jumper arches he tries to form a kind of semi circular ring like structure .. which is ir-regular figure ... making a circle(imaginary) by taking arc (jumper's body) as its part .. gives us a regular figure ie .. a circle but .. its density distribution would be non-uniform ... the body arc part will be high density one .. while other part would be low density (since other part of circle is of air).. so .. centre of mass would be near body & centre of circle .. (but not inside body).. i hope you got it ..
• How to exemplify the center of mass if it is outside the object? For example, if there is a hole in the center of the ruler, how to apply the force 'F'? • That question can be expanded: if the center of Mass is inside an object, how can you apply a force to it (e.g. inside a 3D horse)?
Since the center of mass is the integral (sum) of the objects moments divided by the total mass, you can do the same with applied forces. It's hard to apply a force at a singular point. So sum up all the impact vectors of forces applied to the object and you get a "Center of force". If that matches with the center of mass, you get a rotation free accelleration
• • i would really appreciate if someone would give an accurate answer to this question.
what is the advantage of being able to move the counterweight along the arm? •  