The problem of the mass of a body

In summary, the mass of the sun is assumed to be equal to the total energy of all the particles in the sun. The gravitational binding energy of the sun is not included in this calculation, and may be different than the actual energy of the binding.
  • #1
wLw
40
1
if we have known the density functionρ(r),and then we can calculate the total mass of a spherical body. M=integral of ρ. Now we will say that body has mass M, but I think it is wrong. according to special relativity, mass is equal to energy, so we can also say that body has total energy M,but i think it neglects the gravitational bind energy, which is negative, so the total energy(mass) of that body is smaller than M. and if your solve the Schwa. metric , there is a parameter (here we call Ms), and we define Ms as the mass(or energy) of central body , and I think it includes the energy of binding energy , so Ms is the total energy(mass )of central body, which is not defined by integral ρ, and maybe you can use integral ρ to calculate the mass(energy) of that, but it must larger than Ms. but i have read many papers and books, they all use that integral ρ to represent the mass of body, like a star How is that?、
 
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  • #2
This is A level so you should be able to do the following:
  • What is the mass of the sun?
  • What is the gravitational binding energy of the sun?
  • What is the fractional difference in mass when you consider gravitational binding?
 
  • #3
wLw said:
if we have known the density functionρ(r),and then we can calculate the total mass of a spherical body. M=integral of ρ. Now we will say that body has mass M, but I think it is wrong. according to special relativity, mass is equal to energy, so we can also say that body has total energy M,but i think it neglects the gravitational bind energy, which is negative, so the total energy(mass) of that body is smaller than M. and if your solve the Schwa. metric , there is a parameter (here we call Ms), and we define Ms as the mass(or energy) of central body , and I think it includes the energy of binding energy , so Ms is the total energy(mass )of central body, which is not defined by integral ρ, and maybe you can use integral ρ to calculate the mass(energy) of that, but it must larger than Ms. but i have read many papers and books, they all use that integral ρ to represent the mass of body, like a star How is that?、
If ##\rho## is the mass density then the calculation is correct. If it's a measure of only one aspect of mass, rest mass of the particles perhaps, then the integral will be the total of all the rest masses.

Note that in curved spacetime you will also have to use the correct volume element for your integral.
 
  • #4
wLw said:
I think it includes the energy of binding energy
In general relativity the concept of mass is not defined for all spacetimes. However, for certain specific classes of spacetimes there are a couple of definitions of mass that are used. One is the Komar mass and the other is the ADM mass. See here for an overview of the issues, limitations, and derivations:

https://en.m.wikipedia.org/wiki/Mass_in_general_relativity
 
  • #5
In SR, the mass of a body is usually taken to be E_0/c^2, where E_0 is the total energy of all constituents of the body, in the rest frame of the body.
This energy includes all kinetic and potential energy, as well as \rho_m.
 
  • #6
if I calculate the integral of mass density \rho, it is not the total energy of body, while is just the mass (exclude the binding energy), is it right??
 

1. What is the problem of the mass of a body?

The problem of the mass of a body refers to the difficulty in accurately determining the mass of an object. Mass is a fundamental property of matter that measures the amount of material in an object. However, the measurement of mass can be affected by factors such as the object's composition, density, and gravitational pull, making it a complex and challenging problem for scientists.

2. How is mass different from weight?

Mass and weight are often used interchangeably, but they are actually two different concepts. Mass is a measure of an object's inertia or resistance to acceleration, while weight is a measure of the force of gravity acting on an object. Mass remains constant regardless of location, while weight can vary depending on the strength of the gravitational pull.

3. Why is the accurate measurement of mass important?

The accurate measurement of mass is important in many fields of science, including physics, chemistry, and engineering. It is a crucial factor in understanding the behavior of matter and the laws of motion. Accurate mass measurements also play a critical role in fields such as medicine, where the dosage of medication is based on the mass of the patient.

4. How do scientists measure the mass of a body?

There are several methods that scientists use to measure the mass of a body. One common method is using a balance, which compares the mass of an object to a known mass. Another method is using a scale, which measures the force of gravity on an object and converts it to mass. In advanced scientific research, techniques such as mass spectrometry and nuclear magnetic resonance are used to measure the mass of atoms and molecules.

5. Can the mass of a body ever be precisely determined?

Due to the complexities involved in measuring mass, it is impossible to determine the mass of a body with absolute precision. However, with advanced technology and techniques, scientists are able to achieve incredibly accurate measurements. The most precise measurement of mass to date is the Planck constant, which has a relative uncertainty of only 0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

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