See http://www.feynmanlectures.caltech.edu/I_07.html:

"This statement can be expressed mathematically by the equation

[…] If to this we add the fact that an object responds to a force by accelerating in the direction of the force by an amount that is inversely proportional to the mass of the object, […]"

I wonder the acceleration is not related to the mass of the object. In fact

$$F = ma = \frac{Gmm'}{r^2}\implies a = \frac{Gm'}{r^2}$$

And the famous myth about $g$ is not related to mass (but weight) …

As the statement repeated later (but in the case of holding a string it is a bit different as the Force is related to the mass we throw away), I wonder is it my old age or something wrong here.

## Best Answer

In simple terms the acceleration is proportional to $\frac{1}{\rm mass}$ and the constant of proportionality is the force so $\rm acceleration = \frac{force}{mass}$.

Using Newton's law of gravitational attraction to substitute for the force one gets to the conclusion that in this case (the force being the gravitational force) the acceleration of a mass is independent of the value of the mass as you have shown.

With the string example the origin of the force is different and so if the force is kept constant (independent of the magnitude of the mass at the end of the string) the acceleration of the mass at the end of the string is inversely proportional to the magnitude of the mass at the end of the string.

So the difference between the two examples is that in one the force depends on the magnitude of the mass which is being accelerated whereas in the other the force can be fixed to be any value.