Gravity & Weight: Calculate Total Mass & Weight on a Planet

[ninjagowoowoo]

Q:
The density of a certain planet varies with radial distance as: D(r) = Do*(1-a*r/Ro), where Ro= 6.3096×106 m is the radius of the planet, Do = 3980 kg/m3 is its central density, and a = 0.290. Calculate the total mass of this planet.Calculate the weight of a one kilogram mass located on the surface of the planet.

Can anyone help me out?

 

[whozum]

Just integrate the density function from 0 to the radius.

Once you find the mass, use F = GMm/r^2 to find the gravitational force.

 

[ninjagowoowoo]

ok so i think i integrated wrong, maybe someone can help me out.

First I distributed the Do so:
D(r)=Do - (Do)ar/(Ro)
then i integrated: (everything is constant except for r)

M(r)=-(Do)(a)(r^2)/2(Ro) from zero to Ro what'd i do wrong?

 

[OlderDan]

ok so i think i integrated wrong, maybe someone can help me out.

First I distributed the Do so:
D(r)=Do - (Do)ar/(Ro)
then i integrated: (everything is constant except for r)

M(r)=-(Do)(a)(r^2)/2(Ro) from zero to Ro what'd i do wrong?

Looks like you lost the Do*1 part of the integrand in your integral, but don't forget this is a 3 dimensional object. The mass at any radius is distributed over the surface of a sherical shell of that radius.

 

[squib]

I think I integrated correctly, but am unsure how to convert this to the mass of the planet... multiplying by (4/3)pir^3 does not seem to be an option...

 

[OlderDan]

I think I integrated correctly, but am unsure how to convert this to the mass of the planet... multiplying by (4/3)pir^3 does not seem to be an option...

You are correct aobut (4/3)pir^3 not being an option. That would only apply in the case of uniform density. You need to rethink how much mass there is at some distance r from the center. It depends on the density at that radius, and how much of the planets volume is at that radius, or more correctly stated how much volume is within a distance dr of that radius. You need to think three dimensionally. Where can you go inside the planet without changing your distance from the center?

 

[squib]

I tried putting both those items in the integral, but still no luck... as this gets me a negative answer

 

[OlderDan]

I tried putting both those items in the integral, but still no luck... as this gets me a negative answer

You can only get a negative answer by failing to include the contribution from the Do*1 term in the density.

 

[squib]

K, I'm doing something wrong here, I tried evaluating the integral of (density)*(volume), but that did not get the right answer

 

[BobG]

I think you're not really visualizing just what you're trying to do.

Your density varies with respect to the radius. So what you really have is a bunch of cylindrical shells summed together, each with a different density. Additionally, the height of each cylinder varies.

If I were drawing this out, I'd probably decide it's easier to find the mass of the top half of the sphere using cylindrical shells, then multiply my result by 2 to get the mass of the entire sphere.