# Difference between gravitational and inertial mass

1. May 1, 2013

### chipotleaway

I can't seem to get my head around the difference between the two.
Inertial mass appears in F=ma and is a measure of an object's resistance to acceleration when being acted upom by a force/s. Gravitational mass appears in F=(GmM)/r^2 - what 'role' does mass play here?

2. May 1, 2013

### dev70

there's a huge difference...inertial mass is found from newton's 2nd law..nd it deals when a body moves horizontally..it does not change with speed (provided its speed is very less than that of light)..
whereas gravitational mass is the ratio of weight of the body acceleration due to gravity..as the acceleration due to gravity is not same everywhere on earth..so it changes..

hope that helps..

3. May 1, 2013

### Bandersnatch

Inertial mass resists forces, gravitational mass is a source of forces.

I'm not sure what dev70 meant by saying that gravitational mass changes, though. It doesn't.

4. May 1, 2013

### ehild

Inertial mass is defined by Newton's Second Law: Force causes acceleration of an object, and the ratio of the force and the acceleration is constant for the object.
You can measure also the weight of an object. Imagine you have two objects with the same weight. Their material, shape, size might be different. Apply the same force on both. Will be their acceleration the same? Or will be the weight of two bodies of equal inertial masses the same? The answer is not straightforward. Gravity can act differently on bodies of different material. The equivalence of the two kinds of mass was subject of Eötvös' experiments, for example. http://en.wikipedia.org/wiki/Eötvös_experiment

ehild

5. May 1, 2013

### dev70

well...i actually didnt mean it...mass remains same..but weight varies..that sounds better..thanks for pointing this out..

6. May 1, 2013

### TurtleMeister

This is a source of confusion sometimes because there are actually two types of gravitational mass, active and passive. It's a pet peeve of mine when people use the term loosely without making it clear which one they're talking about. Passive gravitational mass is a measure of strength of a bodies interaction with a gravitational field. Active gravitational mass is a measure of strength of the gravitational field produced by a body. Usually the former is implied when the term "gravitational mass" is used by itself, but not always. And then sometimes people just simply get it wrong.

To give an example of what the differences are:

The inertial mass of your body could be determined by measuring how much force is required to accelerate yourself at a certain rate. Or in other words, Newton's second law.

The passive gravitational mass of your body could be determined by, well, simply standing on your bathroom scales.

The active gravitational mass of your body could be determined by using a torsion balance. And it would be a very delicate and difficult task. That's because the active gravitational mass of your body would be very small.

The equivalence principle tells us that the three methods of determining your body's mass, as described above, will give us the same result (within experimental error of course).

You may ask, how can active gravitational mass be equivalent to the other two types when I stated that it would be a very delicate and difficult task to measure in a human body. The answer is that the equivalence principle, as it applies to active gravitational mass, is a proportional equivalence and not a quantitative one.

7. May 1, 2013

### xAxis

@TurtleMeister
What do you mean by active gravitational mass of your body would be very small. Isn't that mass same as a passive mass, but the field generated by it would be small?

8. May 1, 2013

### TurtleMeister

That's a good question. As I said, gravitational mass is a source of confusion. Sometimes it is described as the mass that produces the field, and sometimes it is described as a measure of the strength of the field itself.

Here is one definition from Wikipedia:
"

I myself just consider it to be the same thing as the standard gravitational parameter, which is defined by Kepler's third law.

9. May 3, 2013

### chipotleaway

Wow that's pretty unintuitive based on what I have learnt so far. I guess I'm getting confusued because I feel like the mass of something determines how much it accelerates in response to a force and that same mass should also dictate the gravitational attraction between it and another mass.

10. May 3, 2013

### TurtleMeister

There are a couple of things in ehild's post that you may have interpreted wrongly.

The equivalence of inertial mass and passive gravitational mass (what ehild is talking about) has been experimentally verified to better than one part in 1013. Proposed satellite experiments will extend that to one part in 1018.

I am assuming that ehild meant for this to be a question and not a statement, since it is followed up with a link to the Eotvos experiments. If it was meant as a statement then it is wrong (based on current theory and experimental evidence).

Last edited: May 3, 2013
11. May 4, 2013

### Popper

As explained above there are three types of mass in Newtonian mechanics:

inertial mass - That which gives movng objects momentum (Weyl's definition of inertial mass)

active gravitational mass - The source of a gravitational field

passive gravitational mass - That on which a gravitational field acts.

It's very important to keep these things in mind and to note the difference between them. When you get into general relativity its essential to know that inertial mass density is different than the density of active gravitational mass while being the same for passive gravitational mass.

12. May 4, 2013

### TurtleMeister

I have not studied general relativity, except for an occasional excursion into the "Special & General Relativity" forum here at PF. Does this difference in mass density between Ma and (Mp, Mi) have a constant proportionality in GR?

13. May 4, 2013

### Popper

No. And it's not always possible to provide a definite quantity in generality which one can call the mass density. Mass is fully described by a tensor.

Using units for which c = 1, and letting the source be an ideal gas then

active gravitational mass density = energy density + 3xPressure

while

inetial mass density = energy density + Pressure

I should point out that different people define these terms differently than above.

14. May 4, 2013

### Staff: Mentor

To further what Popper said, in GR the active gravitational mass essentially doesn't exist. In Newtonian gravity the active gravitational mass is the thing which acts as the source of gravity. In GR, the thing which acts as the source of gravity is the stress-energy tensor.

In a situation where Newtonian gravity is a good approximation to GR then you can make the simplification Popper mentioned.

15. May 5, 2013

### ehild

I talked about classical mass. Each theory has got its own definitions and postulates. Classical physics postulated position, time, and interaction between objects. It also defined velocity and acceleration and Newton set up his laws. The second one is definition of the mass of an object: the inertial mass. There is also the weight of a body: you put a body on the scales and balance it with a standard "weight", then you say that the body has the weight equal to the standard one. That weight is proportional with the gravitational mass, as you balance the force of gravity. As inertial mass and gravitational mass were measured with different procedure, (one dynamic, the other static) it was not sure if they were really proportional or with properly chosen proportionality factor, were they identical. That was proved experimentally with very high accuracy since then, but the concepts themselves were not equivalent.

It was a question, the masses proved to be equal experimentally with high accuracy. So we take them the same when determining acceleration or when we determine the force a big load exerts on a support.
............
You can make different theories on different models. You can make a theory where mass is postulated itself and no distinction exists between gravitational and inertial mass.

When you define a physical quantity you have to make it clear, how is it measured. In SR, rest length (proper length) is measured by comparing it with a metre stick, while you measure length with a different procedure from a moving frame of reference. The different measurement procedures result in different numerical values for the length of the same object.
The same with mass - rest mass is different from the moving mass.

And you can speculate what is mass, if it exist at all? ...

ehild

16. May 5, 2013

### ehild

The distinction between active and passive gravitational mass are attributed to the approximation of the gravitational interaction with the help of "gravitational field". That is a proper approximation when the motion of one of the interacting bodies (the big one) is not influenced by the motion of the "small" body. It is said then, that a gravitational field exist around the big body, determined by its (active) mass. The intensity of this field multiplied by the (passive) mass of the small body is equal to the force acting on the small body, and that force determines its acceleration if divided by the (inertial) mass. That is an approximation which fails when the masses of the bodies are comparable. In case of a pair of twin stars, you can apply the law of gravity to figure out the motion of both starts, but you can not use the gravitational field approximation.

ehild

17. May 5, 2013

### Crazymechanic

to the OP post I could use a gyroscope as an answer and a working model where inertial mass or the physical rotation of a given "mass" wheel counteracts the force exerted on the wheel by gravitational mass , by the way everyone can try to spin a bicycle rim and then try to move it in different angles while it spins.

18. May 5, 2013

### TurtleMeister

@Popper or DaleSpam

Do you mean to say that in a situation where Newtonian gravity is a good approximation to GR that the stress-energy tensor will not necessarily be proportional to Mi and Mp? If this is true then how does the equivalence principle apply to GR?

Classical physics has changed over time. New terms and concepts are incorporated. For example, prior to the twentieth century the concept of mass being divided into the three types that we are discussing, did not exist.1 Or at least there is no written record of it. In your post #4, you stated:

emphasis mine

This refers to the equivalence of inertial mass and passive gravitational mass based on material composition. The experimental verifications that I mentioned are for the torsion balance experiments done by the Eot-Wash Group and the proposed satellite experiment called STEP. These experiments are just more modern versions of the Eotvos experiments that you linked to in your post.

1Even though the concepts did not exist prior to the twentieth century, experiments involving them did, as evidenced by Newton's pendulum experiments using bobs made of various materials.

19. May 21, 2014

### inertian

there is no difference between inertial and gravitational mass; however, how it is measured is the difference

20. May 21, 2014

### DrStupid

Why not using the passive gravitational mass of Earth to measure your active gravitational mass? That could be done by, well, simply standing on your bathroom scales.