The Polearm Master feat reads:
While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.
Let's say my character is wielding a quarterstaff in one hand and a whip in the other hand, and I have the Polearm Master feat. When a character enters a square within 10 feet of me (which is within my reach with the whip), can I use the whip to make an opportunity attack?
It's very understandable that this question got brought up. The original wording of the Polearm Master feat certainly did imply that "your reach" might apply to any weapon, not just your polearm. But more recent reprints of the Player's Handbook (3rd reprint or newer) include new wording of the Polearm Master feat. I've reproduced the 3rd reprint version below (PHB, p. 168, bold and italics added for emphasis: bracketed addition reflects newer reprints than the 3rd reprint one I own).
While you are wielding a glaive, halberd, pike, or quarterstaff [,or spear], other creatures provoke an opportunity attack from you when they enter the reach you have with that weapon.
This change was strangely not mentioned in any errata to date (which do mention the addition of the spear in the list of polearms for this feat), but the newest version of the text has made its way into some official online resources, such as DnD Beyond's PHB compendium content (though it strangely has not been updated on some other DnD Beyond pages, such as the link posted above in this question).
This last italicized phrase makes the situation unambiguous: the word "that" must refer to one of the weapons previously mentioned. So the relevant reach is the reach of the polearm you are wielding, not the reach of some other weapon. Thus, in your example, if you were wielding a quarterstaff and a whip (assume you had the Dual Wielding feat to remove the question of whether you could be said to be "wielding" both weapons if you could only attack with one in a turn), the enemy would provoke an opportunity attack when they moved within 5 feet of you (the quarterstaff's reach), not within 10 feat (the whip's reach). To summarize, for the purpose of the second bullet of the Polearm Master feat, only your polearm's reach matters (not the reach of some additional wielded weapon).
The OA from Polearm Master can only be taken with the listed weapons, by wielding those weapons. If you are hold a quarterstaff and attacking with a whip, you are not wielding the quarterstaff; you are wielding the whip.
The intent is that any OA triggered because you're wielding a polearm is then made with that polearm.
Jeremy Crawford has said:
Q: If I have the war caster feat and a reach weapon can I use a spell instead of the weapon if they move to 15 ft?
A: The intent is that any OA triggered because you're wielding a polearm is then made with that polearm.
So it seems like the designers actually intended for this combination not to work.
In this example, the OA is triggered only because you are weilding a polearm. According to Jeremy, this would mean the intent is that that OA is made with that polearm.
Since the polearm does not have a 10 foot reach and thus cannot make the attack, it would logically imply that opportunity attacks were not intended to be triggered by the non-polearm weapon.
You can decide for yourself if and/or how to apply this guidance at your table.
Let's look at the exact wording of the feat:
While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.
This establishes a condition and an effect if that condition is met. The condition is:
While you are wielding a glaive, halberd, pike, or quarterstaff [...]
That condition is met if you are wielding a quarterstaff and a whip.
The effect is:
[...] other creatures provoke an opportunity attack from you when they enter your reach.
Since the condition is met, you receive that effect, which means that other creatures provoke Opportunity Attacks when they enter your reach. Your reach with a whip is 10 feet, so creatures provoke an OA when they enter that reach.
If WotC didn't want the feat to function in this faction, they would have changed the wording on either the condition or the effect. I've provided two examples of how that might work:
While you are wielding only a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.
This wording would change the condition so that it's no longer met if you're holding any other weapons - it would require you to hold only an eligible polearm to receive the effect.
While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach with that weapon.
This wording would change the mechanics of the effect so that enemies only provoke attacks of opportunity when they enter the reach of the polearm, rather than entering your reach with any weapon.
The writers made a deliberate choice not to restrict the condition (require wielding only a polearm) or the effect (requiring the reach or attack to be with a polearm).
For a similar example, see Crossbow Expert. The feat includes a benefit to ranged spell attacks. By looking at the name of the feat, you might reasonably assume that benefit was only meant to apply to crossbows. However, the writers explicitly chose not to include wording restricting the effect to crossbow attacks, which allows a feat named "Crossbow Expert" to benefit spellcasting. This was confirmed in the Sage Advice Compendium:
Is it intentional that the second benefit of Crossbow Expert helps ranged spell attacks? Yes, it’s intentional. [...] When designing a feat with a narrow use, we consider adding at least one element that can benefit a character more broadly [...]
They intended for the effect to extend to spellcasting, but they didn't explicitly spell it out. Rather, they explicitly avoided restricting the benefit to crossbows only. If they had intended for it to only apply to crossbow attacks, they would have said so.
If it's reasonable for a feat entitled "Crossbow Expert" to benefit ranged spell attacks, it's reasonable for a feat entitled "Polearm Master" to benefit whips.
