Sunday, March 13, 2016

newtonian mechanics - What's the difference between centre of mass & centre of gravity for massive bodies?


My book says:



For most of the small objects, both are same. But for mammoth ones, they are really different ones. And in a gravity-less environment, COG is absent; COM still exists.



Ok, what's the big deal when things are small and big? How these two : centre of mass & centre of gravity?



Answer



Both values are computed as a position weighted average. For the center of mass we average the mass in this way, while for the center of gravity we average the effect of gravity on the body (i.e. the weight).



\begin{align} x_\text{com} &= \dfrac{\int x \, \rho(x) \,\mathrm{d}x}{\int \rho(x) \, \mathrm{d}x} \\ \\ x_\text{cog} &= \frac{\int x \, \rho(x)\, g(x) \,\mathrm{d}x}{\int \rho(x) \,g(x) \,\mathrm{d}x} \end{align}


Now, in the usual Physics 101 "near the surface of the Earth" convention $g(x)$ is constant so these two are equivalent. However, if the body is big enough that we need to account for either the changing strength or changing direction of gravity then they are no longer the same thing.


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