In Newtonian mechanics as already observed and understood by Galieo, inertial and gravitational (heavy) mass are equal, because there is only one form of mass and that is inertial mass as a measure of acceleration vs force per unit of volume. Since Newtonian gravitation is a force per unit of volume, gravitational mass is equal to inertial mass, by definition, as expressed by the fact that the dimension of gravitation is $m/s^2$. See also Chap 18 of Many-Minds Relativity.
Let us compare this insight with what modern physics says as told by Nigel Calder in Magic Universe:
- A succession of experiments to check the equivalence principle—the crucial proposition that everything falls at the same rate—began with Lorand Eötvös in Budapest in 1889. After a century of further effort, physicists had improved on his accuracy by a factor of 10,000. The advent of spaceflight held out the possibility of a further improvement by a factor of a million.
- If another theory of gravity is to replace Einstein’s, the equivalence principle cannot be exactly correct. Even though it’s casually implicit for every high-school student in Newton’s mathematics, Einstein himself thought the equivalence principle deeply mysterious. ‘Mass,’ he wrote, ‘is defined by the resistance that a body opposes to its acceleration (inert mass). It is also measured by the weight of the body (heavy mass). That these two radically different definitions lead to the same value for the mass of a body is, in itself, an astonishing fact.’
- Francis Everitt of Stanford put it more forcibly. ‘In truth, the equivalence principle is the weirdest apparent fact in all of physics,’ he said. ‘Have you noticed that when a physicist calls something a principle, he means something he believes with total conviction but doesn’t in the slightest degree understand.’
- Together with Paul Worden of Stanford and Tim Sumner of Imperial College London, Everitt spent decades prodding space agencies to do something about it. Eventually they got the go-ahead for a satellite called STEP to go into orbit around the Earth in 2007. As a joint US–European project, the Satellite Test of the Equivalence Principle (to unpack the acronym) creates, in effect, a tower of Pisa as big as the Earth. Supersensitive equipment will look for very slight differences in the behaviour of eight test masses made of different materials— niobium, platinum-iridium and beryllium—as they repeatedly fall from one side of the Earth to the other, aboard the satellite.
- ‘The intriguing thing,’ Everitt said, ‘is that this advance brings us into new theoretical territory where there are solid reasons for expecting a breakdown of equivalence. A violation would mean the discovery of a new force of Nature. Alternatively, if equivalence still holds at a part in a billion billion, the theorists who are trying to get beyond Einstein will have some more hard thinking to do.’
Einstein skillfully jumped between definition as a tautology true by construction and physical principle/law, which may be valid/true or not, thereby creating a total confusion. Another aspect is the constancy of the speed of light, which today is used as definition with the meter defined by distance traveled by light in certain time, yet physicists go around and believe that this works because the speed of light is constant. If you cannot distinguish between a definition without content and statement with content, then you may find yourself in trouble and mislead others...
PS This previous post may be consulted: The Principal Difference between Principles and Laws in Physics. Note in particular the distinction that a law is typically expressed as a formula, while a principle is expressed in words e.g. as equality of inertial and gravitational mass.