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

• ninjagowoowoo
In summary: The mass of the entire sphere would be the sum of the masses of the top half of the sphere.In summary, the planet has a varying density, which is due to the fact that it has different-sized cylindrical shells that have different densities. Additionally, the height of the cylinders also varies. The mass of the planet is the sum of the masses of the top half of the sphere.
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?

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.

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?

ninjagowoowoo said:
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.

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...

squib said:
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?

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

squib said:
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.

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

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.

## 1. What is the difference between mass and weight?

Mass is a measure of the amount of matter in an object, while weight is a measure of the force of gravity acting on an object. Mass is a constant property of an object, while weight can vary depending on the strength of gravity.

## 2. How is weight calculated on different planets?

Weight is calculated by multiplying an object's mass by the strength of gravity on that planet. The strength of gravity on a planet is determined by its mass and size. So, an object will weigh more on a planet with a stronger gravitational pull and less on a planet with a weaker gravitational pull.

## 3. How does gravity affect objects of different masses?

Gravity affects all objects in the same way, regardless of their mass. However, objects with greater mass have a greater gravitational force and therefore require more force to move them.

## 4. Can an object have different weights on the same planet?

No, an object will have the same weight on the same planet, as long as the strength of gravity remains constant. However, the object's mass can change, which will affect its weight.

## 5. How does weight differ from mass in terms of units?

Mass is measured in units of kilograms (kg), while weight is measured in units of newtons (N). This is because mass is a measure of the amount of matter, while weight is a measure of the force of gravity, which is measured in newtons.

Replies
7
Views
8K
Replies
3
Views
5K
Replies
2
Views
4K
Replies
11
Views
2K
Replies
9
Views
1K
Replies
9
Views
3K
Replies
4
Views
955
Replies
1
Views
1K
Replies
5
Views
1K
Replies
3
Views
1K