Moon Center of Mass/ Center of Gravity

[schaefera]

Hi,

I was wondering about an issue that we just started in my physics class: we learned that center of mass (CM) and center of gravity (CG) coincide in an object as long as the force of gravity is uniform over that object. So, for something near the Earth's surface we can say that CM and CG coincide because the small change in height means that gravity is essentially uniform over that object; but when you get to the moon, do the CM and CG coincide? It's a very large object, so does gravity have a constant value all along it?

Thanks!

 

[Bill_K]

Both the center of gravity and center of mass of a uniform sphere coincide with the geometrical center. The Earth's pull is not uniform from the front of the moon to the back, but the only effect this has on the moon is to cause a tidal stress.

 

[schaefera]

Is there any reason why this is true? I'm thinking that even if the moon were thought of as nested, spherical shells, that doesn't mean the force of gravity can't be somewhere other than the center of mass (or can it)?

 

[netheril96]

Is there any reason why this is true? I'm thinking that even if the moon were thought of as nested, spherical shells, that doesn't mean the force of gravity can't be somewhere other than the center of mass (or can it)?

I never heard someone referring to "center of gravity" in space. But, conceptually, moon's center of gravity is definitely not center of mass.

 

[Bill_K]

I never heard someone referring to "center of gravity" in space. But, conceptually, moon's center of gravity is definitely not center of mass.

This is a well-known theorem, but if you don't accept it, by all means tell us where you think it is instead. Or see http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/sphshell.html.

 

[netheril96]

This is a well-known theorem, but if you don't accept it, by all means tell us where you think it is instead. Or see http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/sphshell.html.

Yes, the gravitational pull between two uniform spheres can be calculated as if they were two points at their respective center of mass.

But is center of gravity something you need to calculate the magnitude of gravity? No! The "as if gravity exerted on one point" is in the context of torque and rotation!

 

[K^2]

Consider an object consisting of two point masses, m₁ and m₂, separated by displacement vector r. Suppose they are in presence of gravitational field with vector strength g₁ and g₂ respectively. Center of mass is trivially located at r_CM=rm₂/(m₁ + m₂) from m₁.

Finding center of gravity is slightly more complicated. It is a point a combined force from which generates the same amount of torque on the system.

$$\vec{T} = m_2 \vec{r}_2 \times \vec{g}_2$$

$$\vec{F} = m_1 \vec{g}_1 + m_2 \vec{g}_2$$

To generate torque T, force F must be applied to a point:

$$\vec{r}_{CG} = \frac{\vec{F} \times \vec{T}}{||F||^2} + \vec{r}_p$$

Note that the solution is not unique. r_p is any vector perpendicular to F. However, for general g₁ and g₂, there is no solution r_CG = r_CM.