Two identical beakers with the same volume of water are placed on each pan of a double-pan balance. a steel ball is suspended from a string and submerged in the water of one of the containers. a hollow plastic ball of the same volume is submerged in the water of the other container and fastened to the bottom of the beaker by a string. will the balance move, and if so in which direction?

The pan having beakers with steel ball will go down.

Explanation:

Both steel ball and plastic ball will experience buoyant force in upward direction . These forces will have same magnitude because they displace same volume of water. Both of them experience reaction force of buoyant force acting in downward direction which will try to lower the pan equally. But in case of plastic ball which is tied to the bottom , the tense  string will apply a force equal to buoyant force in upward direction on the bottom of the beaker.

This force will balance the reaction buoyant force acting on the bottom of beaker in downward direction . Hence this pan will remain balanced but the other pan will go down.