B Understanding Point Mass: An Example

[Dynamics101]

What does it mean to treat something as a point mass. Give an example as well. Thank you.

 

[Drakkith]

It just means you treat an object as if all of its mass is exactly at a single point. For example, I could model the Earth as being a single point instead of a sphere for the purposes of doing gravitational calculations. We do this because it usually simplifies our calculations a great deal. Note that you can't always do this. If you want to find the gravitational force from the Earth on an asteroid that's 0.5 AU from the Earth, you're probably okay with treating both the Earth and the asteroid as point-masses. If you want to very accurately calculate the variation in the Earth's gravity at different points on its surface, you absolutely cannot treat the Earth as a point mass.

 

[Dynamics101]

It just means you treat an object as if all of its mass is exactly at a single point. For example, I could model the Earth as being a single point instead of a sphere for the purposes of doing gravitational calculations. We do this because it usually simplifies our calculations a great deal. Note that you can't always do this. If you want to find the gravitational force from the Earth on an asteroid that's 0.5 AU from the Earth, you're probably okay with treating both the Earth and the asteroid as point-masses. If you want to very accurately calculate the variation in the Earth's gravity at different points on its surface, you absolutely cannot treat the Earth as a point mass.

So to your last statement, if you don't treat it as a point mass how will you calculate it?

 

[Drakkith]

So to your last statement, if you don't treat it as a point mass how will you calculate it?

No idea.

 

[Dynamics101]

Thank. Last question. What if the mass is evenly distributed.
Would you consider it a point mass?

 

[Drakkith]

Thank. Last question. What if the mass is evenly distributed.

If the object is a perfect sphere, or very, very close to a perfect sphere, then you can treat it as a point-mass in most situations.

 

[Dynamics101]

Thank man!

 

[pixel]

So to your last statement, if you don't treat it as a point mass how will you calculate it?

If the object has some size and is irregular in shape, you have to do an integral over the volume of the object to calculate the quantity of interest. Usually, there are rules of thumb such as: if the ratio of the largest dimension of the object to the distances of interest is below X, then the object can be considered a point mass with an error below Y, where X and Y depend on the specific quantity being calculated.

 

[Nugatory]

Thank. Last question. What if the mass is evenly distributed.
Would you consider it a point mass?

If the mass distribution is spherically symmetrical, then the gravitational field everywhere outside the surface of the object is exactly the same as if the object were a point mass with the mass concentrated at the center. Thus you can consider it a point mass as long as you stay above the surface; this will be the case for planetary motion, the trajectories of dropped objects, ballistics, spacecraft in orbit or free flight, just about all the practically important problems you'll encounter.