Gravitational and inertial mass
We can measure the mass of an object by comparing its weight to the weight of a known object, using a balance. If the known object has a weight of 9.8 Newton, its mass is 1 kg. If the test object also has a weight of 9.8 N, we say its mass is also 1 kg.
We measure the inertial mass of an object by having a force changing the movement of the object when that movement is in a horizontal plane in order to prevent gravity to change the readings. A simple inertial balance is like this: (http://www.arborsci.com/inertial-balance)
The object is placed on a support attached to two metal strips. These strips can move horizontally only when a force pushes it aside from rest position. The movement back and fro depends on the amount of mass on the support: big mass will cause a slow back and fro movement; small mass will cause a fast back and fro movement. If the frequency of different tested mass is plotted on a chart, an unknown mass is used and from the chart its inertial mass can be calculated.
The question asked for years was this: is there a difference between inertial mass and gravitational mass.
With the advent of apparatus that can increase the weight of an object or decreased its weight using an horizontal laminar light beam, a new verification is now possible.
Take two 100 g mass. Each has a weight of 0.98N.
Put one 100 g mass over the apparatus and one under the apparatus. When 100 g mass under the light starts to levitate, put it on an inertial balance. Put the 100 g mass that was over the apparatus with an increased weight on another inertial balance.
Give each balance a small push horizontally and note the frequency of each one. The frequency will be very close to one another even if one has almost triple the weight of the other one.
Put a third mass of 0.98 N on one of the inertial balance and compare the frequency. The frequency of the light mass is almost double.
That will show that the weight of the object is simply a force on the object and does not change the mass of the object.
Conclusion: If 2 objects have the same weight and are both subjected to the same gravity, they will also have the same inertial mass.
We hope that many laboratories will start to make the apparatus to change the weight of objects. In 2016, there was only one of these on planet earth.