Consider an astronaut standing on a weighing scale within a spacecraft. The scale by definition reads the normal force exerted by the scale on the astronaut (or, by Newton's third law, the force exerted on the scale by the astronaut.) By the principle of equivalence, the astronaut can't tell whether the spacecraft is

(a) sitting at rest on the ground in uniform gravity \(g\), or

(b) is in gravity-free space, with uniform acceleration \(a\) numerically equal to the gravity \(g\) in case (a). Show that in one case the measured weight will be proportional to the inertial mass of the astronaut, and in the other case proportional to the astronaut's gravitational mass. So if the principle of equivalence is valid, these two types of mass must have equal magnitudes.