as a school's projector rewinds, one of its reels vibrates. how would you explain this vibration?

Imbalance                                  .

Take it to an auto tire place and have it balanced .

There is either DYNAMIC imbalance, STATIC imbalance, or free space between the axle and the centre of the reel.  Any of these three gives the vibration.  If it reaches a critical speed then OSCILLATION occurs where the magnitude of the vibration increases dramatically.

Static imbalance is where there is more mass on one side than the other F = m w^2 r where m is the amount of the imbalance.  The projector itself is bouncing up and down.

Dynamic imbalance is where the heavy point on one side is not exactly opposite the heavy point on the other side so that in attempting to be thrown out precisely radially the whole reel vibrates from side to side.  The projector vibrates sideways.

The clearance allows the entire wheel to move a little to one side creating the condition for static imbalance ( more mass to one side than the other).

Oscillation means that the wheel is thrown to one side by the force.  It hits that side and bounces off it.  In that time the wheel has turned a half turn so that the force is now moving it in the direction the wheel is bouncing so it hits that side even harder.  Once more exactly 180 degrees later it has bounce to the other side.

At this speed things can get out of control very fast.

The reels and the winding mechanism are not manufactured to very precise tolerances.  They are not perfectly balanced because that would be more costly than warranted (except perhaps for very special, and therefore very expensive equipment).  And the winding itself, at high speed, will probably not be perfect, and imbalance will occur as a result.  Imbalance is not important for projection, which is done at a much slower speed.

That vibrating reel is unbalanced !!!!

It’s not balanced. The film must have something that’s heavier on one side than the other to cause vibrations. 

It’s like a centrifuge.  If it’s not balanced, it will shake. 

Age, need of replacement, maintenance.