Osmosis in cells
Plant cells have a strong cellulose cell wall on the outside of the cell membrane.
This supports the cell and stops it bursting when it gains water by osmosis
- therefore the plant cell will not be permanently damaged by the entry of water.
A plant cell in a dilute solution (higher water potential than the cell contents)
Water enters the cell by osmosis. The cytoplasm pushes against the cell wall and the cell becomes turgid.
A plant cell in a concentrated solution (lower water potential than the cell contents)
Water leaves the cell by osmosis. The cytoplasm pulls away from the cell wall (plasmolysis) and the cell becomes flaccid and the plant wilts. 
The beaker contains water and sugar molecules
Water molecules pass through from solution one to two
In the experiment, eventually the level on the more concentrated side of the membrane rises, while the one on the less concentrated side falls. 
When the concentration is the same on both sides ...
the movement of water molecules will be the same in both directions.
At this point, the net exchange of water is zero and there is no further change in the liquid levels. 
Osmosis in animal cells
Animal cells do not have a cell wall.
- so they change size and shape when put into solutions that are at a different concentration to the cell contents.
For example, red blood cells:
- gain water, swell and burst in a more dilute solution (this is called haemolysis)
- lose water and shrink in a more concentrated solution (they become crenated or wrinkled)
These things do not happen inside the body because ...
... osmoregulation involving the kidneys ensures that the concentration of the blood stays about the same as the concentration of the cell contents. 
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