# Half the size and mass, stronger force of gravity?

• PaulMango
The gravity on the surface of the cube reduces in proportional to the mass you have lost. This would be true of any spherically symmetrical reduction in the mass.

#### PaulMango

I've arrived at a non-sensical solution for a pretty simple thought experiment about gravity. Please let me know where my thinking is breaking down:

Assume we have a cube of even density and a point mass on center of one of the sides. There is a force of gravity between the two of GmM/r^2 where r is half the length of one of the sides (the distance to the center of gravity of the cube). Now imagine the half of the cube farthest from the point mass disappears. It vaporizes. Now the mass of the cube is M/2, but the distance to the center of gravity is r/2. This would indicate an INCREASE in the force of gravity, but this is far from intuitive. Any ideas where I'm wrong?

PaulMango said:
There is a force of gravity between the two of GmM/r^2
This is only true between point-like objects (or outside of spherically symmetric objects).

The shell theorem only applies for spherical distributions of matter. Treating a cube (or half cube) as a point is probably not going to get you great results. :)

Thanks for the reply! I thought that may be the case - didn't realize that equation was based on that assumption.

I'm not convinced the OP's question has been addressed.

I suspect that his question will still exist, even he switches to using a spherical mass. I guess that's really up to the OP though.

PaulMango said:
I've arrived at a non-sensical solution for a pretty simple thought experiment about gravity. Please let me know where my thinking is breaking down:

Assume we have a cube of even density and a point mass on center of one of the sides. There is a force of gravity between the two of GmM/r^2 where r is half the length of one of the sides (the distance to the center of gravity of the cube). Now imagine the half of the cube farthest from the point mass disappears. It vaporizes. Now the mass of the cube is M/2, but the distance to the center of gravity is r/2. This would indicate an INCREASE in the force of gravity, but this is far from intuitive. Any ideas where I'm wrong?

Your fundamental mistake (apart from using a cube rather vthan a sphere) is to assume that the centre of gravity of a hollow cube or sphere is not at its centre. In this case, a point on the surface is still a distance r from the centre of mass.

So, the gravity on the surface reduces in proportional to the mass you have lost. This would be true of any spherically symmetrical reduction in the mass.

If he uses a spherical masses instead he will need to specify what parts of the sphere disappears. If half the sphere mass disappears in such a way that the result is a new sphere, the force will decrease if applying the spherical solution. (Naturally, it will always decrease if using a correct approach.)

At the risk of re-hashing what has already been said. If you want to compare apples with apples then the smaller cube must still be a cube. It must have been cut in half in all three dimensions. So you slice away the far 1/2 cube. Then you slice off the outside of the flattened shape that remains. Result is a cube that is 1/8 the mass of the original and is at 1/2 the distance. An argument from symmetry indicates that it must therefore have 1/2 the attractive force (at its surface).

The gravitational attraction of a cube of uniform density at its surface scales directly with the linear dimensions of the cube, just like the gravitational attraction of a sphere of uniform density scales directly with the linear dimensions of the sphere.

PaulMango said:
I've arrived at a non-sensical solution for a pretty simple thought experiment about gravity. Please let me know where my thinking is breaking down:

Assume we have a cube of even density and a point mass on center of one of the sides. There is a force of gravity between the two of GmM/r^2 where r is half the length of one of the sides (the distance to the center of gravity of the cube). Now imagine the half of the cube farthest from the point mass disappears. It vaporizes. Now the mass of the cube is M/2, but the distance to the center of gravity is r/2. This would indicate an INCREASE in the force of gravity, but this is far from intuitive. Any ideas where I'm wrong?

I am pretty sure that the new r is not r/2, but something greater than that (although less than the original r).

The decrease in M is much greater than the decrease in r. Meaning that gravity will decrease.

If it was just sliced in half would be less gravity to the point on center face because y and z extremities cancel out, the point is closer to the center of mass so to speak.

## 1. What does it mean for an object to have "half the size and mass, stronger force of gravity"?

It means that the object is physically smaller and has less mass compared to another object, but its gravitational pull is stronger due to its density.

## 2. How is this possible? Isn't gravity directly related to mass?

Yes, gravity is directly related to mass, but it is also affected by an object's size and density. An object that is smaller but has a denser mass will have a stronger gravitational force than a larger object with the same mass.

## 3. Can you give an example of an object that has "half the size and mass, stronger force of gravity"?

One example is a neutron star, which is about the size of a city but has a mass equivalent to that of our sun. Due to its extreme density, it has a much stronger gravitational force compared to the sun.

## 4. How is this concept relevant to space exploration?

Understanding the relationship between size, mass, and gravity is crucial for space exploration. It helps scientists determine the strength of a planet's gravitational pull and how objects will behave in that environment.

## 5. Can this concept be applied to everyday objects on Earth?

Yes, this concept can also be applied to everyday objects on Earth. For example, if you have two objects with the same mass but different densities, the one with the higher density will have a stronger gravitational pull.