What does the inertial balance compare or measure?

An inertial balance is used to measure the value of the inertial mass. It can also be used to compare the gravitational masses. 

Inertial balance is basically a spring device that vibrates the sample which is being measured.
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Which of the two masses is greater-inertial mass or gravitational mass?

Inertial mass and gravitational mass are precisely equal to each other. The equivalence of these masses makes it possible for all the objects to fall at the same rate.
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What is the working principle of common balance?


Common balance is the balance which has each arm suspended.The unknown mass is placed in one arm and the known mass in another until they both become equal. 
Therefore this balance works on the principle of moment of weights. When the weights become balanced, equilibrium is attained. 
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How do the inertial and gravitational masses differ - quantitatively or qualitatively.

The inertial mass and gravitational mass differ qualitatively.

Inertial mass is found out using Newton's law. On applying a force of F Newton to an object, a being the acceleration we will get the inertial mass as m= Fa.

Graviational mass is defined by the gravitational force which states that there is gravitational force between any two pair of objects. 
F = Gm1m2r2
where, m1 and m2 are the masses of the objects. 
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What docs the common balance compare or measure?

The common balance measures or compares the gravitational masses.
The arms of the balance move up and down, leaving the only position for equilibrium as exactly straight, for equal masses. 
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