Lever and Acceleration: Greater Mass or Greater Accel?

[ultrauser]

If I have 10m lever with a center of the mass 2m from the right side it must mean the shorter arm have greater mass to keep balance. Does it mean longer (lighter) arm have greater gravitational acceleration cause it must travel greater distance in the same time ?

 

[Simon Bridge]

If I have 10m lever with a center of the mass 2m from the right side it must mean the shorter arm have greater mass to keep balance.

Only if it is in balance - a lever does not have to balance.

Does it mean longer (lighter) arm have greater gravitational acceleration cause it must travel greater distance in the same time ?

No.
1. the acceleration of gravity is the same for all masses anyway
2. since the two sides are in balance, the force due to gravity on both sides is the same - it cannot be accelerating: so you have a contradiction in your statement.

Notes:
In order to accelerate the lever, you need an additional unbalanced torque.
That torque causes the same angular acceleration in both arms.
The linear/tangential acceleration varies along the length of each arm - and is the same (with opposite sign) at equal lengths. The difference in linear accelerations of the endpoints is due to geometry.