Earth rotation discrepancy

Some time ago, I discovered that planet earth's sidereal rotation period equals 23 hours, 56 minutes, and 4.1 seconds (see for example this Wikipedia article). The sidereal rotation is the rotation of a celestial object around its axis. Thus, earth takes a little less than 24 hours to rotate around its own axis.

Now I wondered how it could be that our clocks measure 24 hours for one day instead of only 23 hours, 56 minutes, and 4.1 seconds. Though the difference seems quite small, we just need to do some basic maths to uncover the ominous effects of that situation:

• after 10 days, the difference between clocks and earth rotation equals ~39,5 minutes
• after 30 days, the difference equals almost two hours
• after half a year, midnight would be around noon

As you can see, there must be some flaw in this logic. We would most probably have realized if at midnight the sun would be at the highest point on the sky.

So, what do you think is the reason that our clocks work fine, although earth seems to rotate a little bit too fast?

In short, in a year, the earth is making "an extra rotation" due to Coin Rotation Paradox. Therefore it adds $$24/365$$ hours which is roughly $$4$$ minutes so that the midnight is still midnight~