C172 flight instructor reported noticing exhaust pipe was missing after landing.

Narrative

After we landed and were parking the airplane; I (the CFI) noticed half of the exhaust pipe was missing. When we were preflighting the plane; both me and my student observed there were a few small holes on one side of the exhaust pipe. I gave it a couple tugs back and forth to make sure everything was in place and not moving or wiggling. There was no movement or other indication that it was about to fall; so we decided to continue with the flight. We took off; did a couple turns around a point; and came back to land. The entire flight weather was great; very smooth surface-2500' MSL; but as we were on the 45 for right downwind rwy XX; around 2000' MSL and 4 miles north of the airport; we encountered sudden moderate turbulence that lasted for about 5 secs. At the time the student was flying; and after noticing that they became anxious; I took the flight controls and gave them some time to breathe and relax. Once they said they was good to fly again; and noticing the air was very smooth again; I gave them control of the plane. Air continued to be very smooth; until on downwind; when we encountered moderate turbulence again. At this point; I took back the controls of the plane; and told my student I will land; just to make sure I was ready to react in case we would encounter more turbulence out of nowhere. I was beginning to suspect some low level wind shear; but there were no issues landing or controlling the plane on base and final; and neither me nor the student heard or felt anything else that would indicate anything separating/hitting the plane. I believe the piece could have separated on one of those two occasions when we encountered the turbulence. Moving forward; I am more aware of the mechanical effects of turbulence or strong gusts on the parts of the plane. Especially with holes or missing screws; that part of equipment loses some of the mechanical strength; and weather could give it the final blow. If I will encounter a similar situation in the future; I will postpone or cancel the flight until maintenance checks or replaces the equipment.