Can we replace 'of' with 'on' in the following construction?

Are the following two constructions substitutes? In other words, can they be used as an alternative to each other?

• The boy broke the knob of the door.
• The boy broke the knob on the door.

If the answer is 'Yes', would there be any difference in meaning?

Solution 1:

“The boy broke the knob of the door.” All we know is that the knob belonging to the door was broken by the boy. We do not know if the knob was attached to the door or not. It may have been on the door or it may have been removed for repair, or was being prepared for installation.

“The boy broke the knob on the door.” This is ambiguous. He may have struck a hard door with a delicate knob - perhaps ceramic, thus breaking the knob. Or he may have broken the knob (that was) on the door while the knob was attached to (=on) the door.

In conclusion, your statements are not necessarily saying the same thing.

Other prepositions such as off or from pose the same problems. For example: “The boy broke the knob off the door.” This is ambiguous. He may have forcibly removed (broken off) the knob. Or he may have broken a knob that was already off the door. This use of off the door as synonymous with of the door is idiomatic.