Is "Mass" really measuring unit for inertia?

Question

Imagine if two objects of identical mass are under two different gravitational field,let's say two different planets (with Different value of gravity) both of the objects are of same mass,but we can easily notice that one body will be easier to move as compared to another (the object which will be on the planet with less gravity will move easily [obviously]), but the mass of both objects is identical, as mentioned above, which means that the inertia of both of the objects should be equal, but one body will be easier to move than the other, which means both of them have different inertia (as inertia is property often body to resist in change in motion).

So does this mean that weight is measure for inertia rather than mass being the unit to measure inertia.

I would like to mention the fact that this problem was also highlighted by Richard P. Feynman but I was not able to find its appropriate solution anywhere.

Edit: I removed 'sir' before name of Feynman because I never knew that sir is added only to the title for people who received knighthood.)(This went off-topic)

Answer 1

Imagine a 10kg curling stone on a flat ice surface on Earth. If we apply 10N of horizontal force, the stone will accelerate at about 1 meter per second per second. On the Earth, a 10kg stone weighs approximately 98N.

Now imagine the same 10kg stone on a flat ice surface on the Moon. If we apply 10N of horizontal force in this scenario, the stone will still accelerate at about 1 meter per second per second. On the Moon, a 10kg stone weighs approximately 16N.

As you can see, the inertia of the stone is the same in both cases, but the weight of the stone is very different. This shows that it is the mass, not the weight, that is the appropriate unit of inertia.

(There are two reasons your intuition tells you that heavier gravity will make it harder to move a weight; one is that when you are carrying an object, you have to lift it against the force of gravity, and the other is that when you are pushing an object the heavier it is the greater the force of friction has to be overcome. But in both cases this is because there are other forces involved, not because of inertia. In the example given above, we are dealing with horizontal motion on a surface with very little friction, so to a good approximation no other forces are involved.)

Answer 2

Physicists distinguish gravitational mass from inertial mass. In practice we find that gravitational mass is equal to inertial mass, but the distinction is important because conceptually they need not be the same.

A measurement of weight is, in effect, a measurement of gravitational mass. That is to say, the amount of gravitational force acting on a body as a result of a gravitational field.

A measurement of inertial mass would mean a measurement of the acceleration resulting from a known force.

While these concepts are distinct, it is not clear to me that we could have a theory of gravity consistent with observation in which gravitational mass is not equal to inertial mass.

Answer 3

So does this mean that weight is measure for inertia rather than mass being the unit to measure inertia.

No. Inertia is resistance to change in velocity (acceleration, $a$). From Newton's second law

$$a=\frac{F_{net}}{m}$$

where $F_{net}$ is the net force acting on the mass $m$..

It's true that a mass $m$ will be harder to accelerate upward in opposition to the direction of the force of gravity on a more massive planet to get the same acceleration, but that's because a greater upward force has to be applied to obtain the same net force and thus the same acceleration.

The inertia of the mass $m$ on all planets is the same because the same net force is required to produce the same acceleration on all planets, or anywhere in space for that matter.

Hope this helps.

Answer 4

I remember reading one of Arthur Clarke's books years ago where he pointed out the misconception that massive objects (which would be heavy on Earth) would be easy to move around in the weightlessness of space. They still have mass and therefore inertia.

Answer 5

we can easily notice that one body will be easier to move as compared to another

This is the error in your reasoning. The low-gravity object will only be easier to move in two situations:

• Moving it against the gravitational potential (e.g. lifting it)
• Moving it against friction that is proportional to the gravitational force (e.g. sliding it across the planet's surface)

The basic equation for inertia is $F = ma$. As the masses are equal, the same acceleration will always require the same net force. If you minimise the two forces working against you above (e.g. roll a ball of ice horizontally across a frozen lake) then both objects will be exactly as easy to move as each other.

Answer 6

Expanding on Harry Johnston's answer, if you had a fairly large, nice round pebble weighing 1kg and held it your hand while standing on Earth, it would exert 9.8N force and feel about the same as an everyday bag of flower. If you threw that stone, you would expend effort and feel a force against your palm as you accelerated the stone.

If you then flew up to the moon and held that pebble in your hand, it would be feel like you were holding only about 1/6 of a bag of flower. However, if you threw it exactly the same way as before on Earth, you would feel the same force against your palm because your muscles would be exerting the same force against the same mass.

Of course, the stone would travel further on the moon, because it is accelerated more slowly towards the ground.

Answer 7

I would like to take a perspective that has not been highlighted yet, I suppose it is a bit unconventional in this context but is appropriate nonetheless in my opinion.

Mass is a measure of inertia as measured in an inertial frame, i.e., the frame in which an object free of the influence of external forces continues to move at a constant velocity. In the presence of gravity, the inertial frame is really the freely falling frame. Thus, on the surface of the earth, it would be a frame moving at $9.8\text{ m/s}^2$ towards the surface of the earth as observed by a frame attached to the surface of the earth. On the moon, this would be a frame moving at $\frac{9.8}{6}\text{ m/s}^2$ moving towards the surface of the moon as observed by a frame attached to the surface of the moon. Now, if you take an object of the same mass in both these freely falling frames, you would indeed require the same amount of force to accelerate it to a certain acceleration, because the effect of gravity would have been canceled out (you don't feel the gravitational force in a freely falling frame, e.g., astronauts in the ISS float).

Answer 8

In a nutshell: weight is the application of gravity on mass.

Under a lack of gravity (or under micro-gravity conditions), objects still have inertia.

Inertia cannot depend on gravity.

Answer 9

You can attribute properties to things for e.g. colour, smell, a name etc. One such property is the ratio of applied net force $\vec F$ to induced acceleration $\vec a$. Like other properties, you don't expect the ratio to be independent of almost everything: it could depend on the material, the place of doing the experiment, surroundings, the temperature who knows? What does one even mean by the ratio of two vectors - it mayn't even be a scalar.

Turns out that the ratio is remarkably independent of the object's other properties its calculated for. It doesn't depend on the type of element the object is made up of*, nor its temperature** nor where the object is. Moreover, there are frames of reference in which its completely characterised by a corresponding single scalar number for every object.

This property is called mass.

Human beings have developed a sense of measuring mass in the form of inertia. You push and see if something moves. The harder the push, more the inertia. The ratio discussed above measures precisely that: amount of push per unit movement. So inertia is commensurate with mass.

Point to note here is that one has to push to get a sense of inertia. People can push stuff over all kinds of things: push a cart in a meadow, push a car on a highway, push yourself on a skating rink, push yourself on dry sandpaper. Turns out there is a different inertia to things depending on the surface. So is the mass different?

"No, No", says the experimentalist. We blame the variation on extraneous circumstances and label that as friction. Its the surface to blame not the object.

You see, the sense of inertia that one has is not as controlled of a scientific property as the mass. If one measures the ratio, far, far away from anything and everything (don't ask how), one would find that its just one scalar $m$.

Its therefore easy to think, as you say, that obviously things would be harder to move around on a planet with stronger gravity. That is what intuition would say, developed on a planet with just one gravity.

But you would be wrong. You see, its again not the mass which is making things difficult here. Your sense of inertia is off because in the mental picture that you have, gravity is acting invisibly to make life harder for you.

Saying moving a block around is harder in stronger gravity and so its got more inertia is like saying a car stuck in dried concrete is heavier. Yeah its harder to push, but it is still the same heavy - the same mass. Its just being held in place very strongly. You don't say that it's heavier or its got more inertia: you just say its being held down firmly.

... After pushing a block around in a stronger gravitational field, you'd probably be (very) tired. So you lay down the block and go to sleep. And then you try to lift it up in the morning.

Human beings have developed a sense of inertia when things that have been laid down are picked back up. We call it weight. Since lifting is just like pulling(pushing)-just in some other direction, weight feels like inertia to us. If something weighs more it most certainly has more inertia and therefore more mass, so we feel.

And that is the root of all confusion.

You see, unlike mass which has the remarkable property that its independent of the amount^ of the applied force, weight doesn't.

In fact one can make weight zero. While lifting such things one wouldn't have to apply any force at all. To push(accelerate) them though, one would. So the inertia would exist even without the weight. Associating inertia with mass therefore makes more sense than weight.

Alas most of the earthlings are earth bound, where they can't really change their weight without varying their mass, so intuitively they would always be the same to us and associating inertia with either would do no harm. Astronauts would beg to differ.

In short , things are harder to in stronger gravity because they weigh more not because the have more inertia.

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*In the sense that you can have the same mass to be made up from any elements. Changing the elements will, of course change mass.

**non-relativistic

^Even more remarkably it doesn't depend on the nature of the applied force.

Answer 10

It's an old question, but let my add one more aspect to the answers.

First of all, intuition fails when talking about lower or higher gravity, as most of us don't have any experience with gravities different from earth, so "obvious" things can easily be wrong.

Thought experiment

There are the so-called zero-G flights where an airplane following a parabolic flight curve creates a no-gravity environment. Now imagine someone shooting a gun during this flight (thought experiment only, please!).

The destruction caused by the bullet comes from its inertia, as you'll surely agree. And you'd surely expect the bullet to cause the same damage as in normal flight.

But, according to your theory of inertia depending on the weight, there shouldn't be any inertia, meaning no damage, because in zero-G, the weight of oll objects, including the bullet, is zero. So, either the expected outcome of our thought experiment is wrong, or your theory is wrong.

P.S. If you want to do a real zero-G experiment, please do not use a gun, but something harmless. And of course, it should (hopefully) be impossible to smuggle a gun on board of an airplane.

Answer 11

One interesting aspect is that when you accelerate a body against a gravitational field, for example by starting a rocket, you actually accelerate the gravitationally bound bodies as well! ¹ No wonder it's hard — you are dragging Earth behind you! ;-)

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_{¹ The "starting rocket" scenario is on closer examination a bit complicated because a substantial, in fact dominating fraction of its mass (the fuel and oxygen) is actually accelerated towards the ground; the movement of the rocket's center of the original overall mass (including fuel and oxygen) is actually towards Earth during a start. Additionally, the exhaust hits the ground during the early phase and creates a force on the Earth which is close to the rocket's weight. In order to avoid all these real life complications it would be best to imagine an infinitely long, mass-less tether which lifts an object from the ground. Please don't ask what it is suspended from. The second best scenario is an accelerating satellite with a photon or at least ion drive, involving much less reaction mass.}