Weight and Inertia (are they related?)

Question

R. Feynman wrote in his lecture (The Feynman Lectures on Physics: Chapter-9)

"Weight and inertia are proportional, and on the earth’s surface are often taken to be numerically equal, which causes a certain confusion to the student. On Mars, weights would be different but the amount of force needed to overcome inertia would be the same."

I am not able to understand how weight and inertia are related because everywhere on internet, everyone is only showing the relation between inertia and mass.

So, is their any relation between them(if so please explain...) or is he wrong in his statement.

Answer 1

Weight is the measure of how much force does gravitation exert on an object. It can be expressed in units of mass, by dividing by the Earth's standard gravity: $$\text{weight} = \frac{\text{gravitational force}}{9.80665\, m/s^2}$$

Inertia, or inertial mass, is a measure of how much force do you need to apply to an object to cause acceleration: $$ \text{acceleration} = \frac{\text{force}}{\text{inertial mass}}$$

It turns out that the inertial mass also happens to affect how big the gravitational force acting on an object is: $$\text{gravitational force} = (\text{inertial mass})\cdot(\text{intensity of gravitational field}) $$ but that's not a part of the definiton of inertial mass, that just happens to be a law of physics.

We have then $$\text{weight} = (\text{inertial mass})\cdot\frac{(\text{intensity of gravitational field})}{9.80665\, m/s^2} $$ If the object is affected by gravitational field of intensity equal to Earth's standard gravity, it will have weight equal to the inertial mass. This is however not true when the intensity of gravitational field is different, for example on Mars, and even on Earth the gravitational field varies from point to point.

Answer 2

Weight and inertia are proportional to each other with the constant of proportionality being the strength of the local gravitational field or, equivalently, the absolute value of the acceleration that the frame of reference that the object is being measured in is undergoing.

So, on the surface of the Earth, there is a gravitational field whose absolute value is approximately constant, and is given by $g \approx 9.8\,\mathrm{ms^{-2}}$. This means that a mass of $1\,\mathrm{kg}$ has a weight of approximately $9.8\,\mathrm{kg\,m\,s^{-2}} = 9.8\,\mathrm{N}$.

Because of this approximate constancy of $g$ and the fact that almost everyone lives on the surface of the Earth, we often become lazy and talk about something 'weighing $n\,\mathrm{kg}$', and we have scales which read in kilograms or grams, or pounds or ounces, when in fact what these scales are doing is measuring force, not mass: on the Moon, all these scales would be wrong, because the constant is different there ($g_\text{moon} \approx 1.6\,\mathrm{ms^{-2}}$).

Inertia and mass, or inertial mass and gravitational mass, are always (we think) the same, and this is very important for theories of gravity.

Answer 3

I think what he's saying is that an object that is more massive (or weighty) has more mass and hence more inertia.

Actually, we must change the words from “light” and “heavy” to less massive and more massive, because there is a difference to be understood between the weight of an object and its inertia. (How hard it is to get it going is one thing, and how much it weighs is something else.) Weight and inertia are proportional, and on the earth’s surface are often taken to be numerically equal, which causes a certain confusion to the student. On Mars, weights would be different but the amount of force needed to overcome inertia would be the same.

As you can see that he clearly differentiates between the terms weight and inertia. What he's trying to mean is that in Mars too, an object that is more massive will have more weight than an object that is less massive.

However if you consider a body of some mass $m$ and perform your measurements in different gravitational fields then you'll get different values of weights but that doesn't mean that the inertia has changed. Inertia is a measure of an object's resistance to changes in its state of motion or rest which is equal to the mass of the body or the amount of matter it has.

In an uniform gravitational field the inertia of a body is directly proportional to its weight.

Answer 4

While reading the same chapter I got a bit confused with that same part:

"Weight and inertia are proportional, and on the earth’s surface are often taken to be numerically equal, which causes a certain confusion to the student"

Actually, in the recording, Feynman never said that weight and inertia are numerically equal, so my guess is that it was a mistake in the transcription. What Feynman said is that inertia and weight are proportional quantities on the Earth's surface. If you have two objects, one heavier and one lighter, you will find that the heavier one has more inertia and the lighter one has less inertia. But, that is ONLY in a portion of space with a constant gravitational acceleration, such as in the Earth, with an acceleration 'g'. What does the latter mean? It means that if you take the heavier object to a portion of space with less gravity, such as the moon (with a gravity of 1.62 m/s/s), it will feel lighter, but that does not mean that it has less inertia, it will have the same inertia as in the Earth. That is, it will have the same tendency to resist changes in velocity as in the Earth. Why is that so? Why is inertia proportional to the weight in a constant gravitational field?

Inertia is a property that describes the tendency to resist changes in motion (in velocity, and therefore, to resist acceleration). That tendency, as you are familiar, depends on mass (more precisely, inertial mass, which has the same value as gravitational mass, so they can be used interchangeably). Therefore, although mass is usually defined as an amount of matter, it can also be defined more abstractly as a measure of inertia (as a measure of the tendency to resist changes in velocity). Therefore, since mass is proportional to the weight (the more mass, the more weight an object), and since mass is a measure of inertia, it follows that inertia is proportional to the weight. But remember that the latter is only true in a constant gravitational field.

The takeaway is the following: Inertia depends on the inertial mass of an object, and therefore, regardless of your position in a gravitational field (in space), your inertia will always be the same, because the amount of matter you hold is the same. However, weight, which also depends on mass (gravitational mass, the amount of mass that interacts with gravity; which remember that according to experiments, is equal to the inertial mass) can change depending on your position in a gravitational field, even if your mass is the same.