What is the relationship between Gravity and Inertia? Einstein told us that gravity and inertia are identical. And from the fact that two different masses fall at the same rate, I believe we can say that gravity and inertia are equal (That is, the inertia of a dropped larger mass is exactly sufficient to slow it’s acceleration to the same level as a dropped smaller mass, regardless of them being dropped on the Earth or on the Moon). But is this where we are left hanging: that gravity and inertia are both identical and equal? Is gravity inertia? Or is inertia gravity? What is the next step beyond saying that gravity and inertia are both identical and equal?
[Physics] the relationship between gravity and inertia
general-relativitygravityinertia
Related Solutions
Moment of inertia
The definition of (mass) moment of inertia of a point mass is
$$
I=r^2m
$$
However in the real world you don't encounter point masses, but objects with non-zero volume (finite density). And leads to an integral to determine moment of inertia
$$
I=\int_m{r^2dm}=\int_V{\rho(r)r^2dV}=\int_x{\int_y{\int_z{\rho(x,y,z)(x^2+y^2+z^2)dz}dy}dx}
$$
The solutions of this integral of a few bodies, with constant non-zero density within geometric volume and zero density outside of it, can be found here. For example the moment of inertia of thin rod rotating around its center of mass is equal to $I=\frac{mL^2}{12}$ and for a solid cylinder $I=\frac{mL^2}{2}$.
Experimental setup
In your experimental a string is on one end connected to the hanging mass, lead over the pulley and then wounded around a drum (the other end is also connected to the drum). This drum is of the object from which you would like to determine its moment of inertia and it is assumed that it can rotate freely (without slip) around its axis.
According to your documentation you measure how far the pulley has rotated, I will call this angle $\theta$, and its first and second time derivative $\omega=\dot{\theta}$ and $\alpha=\ddot{\theta}$.
The displacement of the hanging mass is related to the angular displacement of the pulley and its radius, $r_p$, assuming that the string does not slip, so
$$
s=r_p\theta
$$
where $s$ is the vertical downward displacement of the hanging mass.
This displacement is equal to the amount of string unrolled from the drum (assuming that the string is not elastic), which means that the angular displacement of the object from which you would like to determine the moment of inertia, I will call this $\theta_I$, can be calculated from this the other way around using the radius of the drum $r_d$
$$
s=r_p\theta=r_d\theta_I\rightarrow\theta_I=\frac{s}{r_d}=\frac{r_p}{r_d}\theta
$$
This linear correlation also applies to the $\omega$ and $\alpha$.
The only force applied on this system (which can perform work) is gravity on the hanging mass. Using all this you can derive the equation of motion (possibly using free body diagrams and tension in the string).
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.)
Best Answer
Yes, Einstein did say that gravity and inertia are identical, despite people in the comments telling you to the contrary. This is a common error derived partly by Einstein’s equating of gravitational mass with inertial mass (in his principle of equivalence), but mostly simply because gravity and acceleration look like different phenomenon.
You could say that gravity and inertia are identical, and that the gravitational field and acceleration are inductive pairs (similar to the electromagnetic field and electric current.) A gravitational field induces acceleration, and acceleration induces a gravitational field.
From Einstein’s 1918 paper: On the Foundations of the General Theory of Relativity… http://einsteinpapers.press.princeton.edu/vol7-trans/49
“Inertia and gravity are phenomena identical in nature.” - Albert Einstein
In a letter Einstein wrote in reply to Reichenbacher... .http://einsteinpapers.press.princeton.edu/vol7-trans/220
“I now turn to the objections against the relativistic theory of the gravitational field. Here, Herr Reichenbacher first of all forgets the decisive argument, namely, that the numerical equality of inertial and gravitational mass must be traced to an equality of essence. It is well known that the principle of equivalence accomplishes just that. He (like Herr Kottler) raises the objection against the principle of equivalence that gravitational fields for finite space-time domains in general cannot be transformed away. He fails to see that this is of no importance whatsoever. What is important is only that one is justified at any instant and at will (depending upon the choice of a system of reference) to explain the mechanical behavior of a material point either by gravitation or by inertia. More is not needed; to achieve the essential equivalence of inertia and gravitation it is not necessary that the mechanical behavior of two or more masses must be explainable as a mere effect of inertia by the same choice of coordinates. After all, nobody denies, for example, that the theory of special relativity does justice to the nature of uniform motion, even though it cannot transform all acceleration-free bodies together to a state of rest by one and the same choice of coordinates.” - Albert Einstein
From Albert Einstein’s book: The Meaning of Relativity, pg 58
“…In fact, through this conception we arrive at the unity of the nature of inertia and gravitation. For according to our way of looking at it, the same masses may appear to be either under the action of inertia alone (with respect to K) or under the combined action of inertia and gravitation (with respect to K’). The possibility of explaining the numerical equality of inertia and gravitation by the unity of their nature gives to the general theory of relativity, according to my conviction, such a superiority over the conceptions of classical mechanics, that all the difficulties encountered must be considered as small in comparison with the progress.” - Albert Einstein
Here and in other places Einstein specifically emphasizes the equivalence of gravity and inertia, and not merely the equivalence of gravitational and inertial mass.
Yes, that is kind of where we are left hanging.
The next step would be in solving in greater detail the physics of inertia. You can search for things like “source of inertia” to get an idea of how some physicists in the past have approached this problem. My feeling is that when the mystery of inertia is more or less solved, Einstein’s assertion on the equivalence of gravity and inertia will be validated.