Since mass is a measure of the inertia of a body we might reasonably expect
the moment of inertia of a body to depend on the mass. 




For example, imagine two flywheels like the one shown here, one made of
aluminium (density about 2700kgm^{3})
and the other of lead, density about 11300kgm^{3}). 

(A "flywheel" is just a wheel but usually
quite a massive one, used for storing energy by virtue of its rotation.) 

Which one would be harder to speed up or slow down? 

The answer is obvious. 



Experiments show that, if we compare bodies of similar shape and size
but having different masses, the moment of inertia,
I is directly proportional to the
mass. 

(This also assumes we are rotating the bodies around the same
axis.) 






Now consider a flywheel like the one shown here. 

Imagine that it has the same total mass as the one above. 



It is clear that here the mass is
much further away from the axle than before. 



This means that, for a
given angular speed, ω, the linear speed
of the particles of which the wheel is made is much greater (so they
possess more kinetic energy). 



Therefore, to accelerate the wheel from zero to
ω would require more work to be done
and
so we reasonably would expect greater forces and/or torques to be needed. 



We conclude that this wheel has a greater moment of inertia
the
the previous one. 



In conclusion: 

The moment of inertia of a body is directly proportional to its mass
and increases as the mass is moved further from the axis
of rotation. 



You can convince yourself of the second part of the conclusion next time you
buy something heavy (massive) in a supermarket, eg a pack of water bottles
(or, maybe more interestingly, beer or wine bottles). 

Place the massive packs near to yourself in the supermarket
trolley... no problem when turning to avoid a collision with another
trolley... 



Now try putting the pack at the front end of the trolley, furthest from
you... you will find the risk of collisions much greater! 



N.B. 

The fact that I depends on mass
distribution means that the same body can have different moments of
inertia depending on which axis of rotation we consider. 
