[clever story omitted]
So, just how much would on cup of protons weigh? In pounds, please.
It wouldn’t hold together, but assuming it did, you need to provide the exact volume of the cup.
Well a neutron star is basically 3.7x10[sup]17[/sup]kg/m[sup]3[/sup] and a cup is basically 250 ml…so around 9.2x10[sup]13[/sup]kg…or 20x10[sup]13[/sup]pounds.
And I’m using a neutron star since a collection of protons will fly apart due to electrostatic repulsion.
I don’t think this is answerable, since protons have varying densities depending on where they’re found. Hydrogen (a proton with an electron on it) is .08 g/L at standard Temperature and pressure.
.036 lbs - assuming liquid form and earths gravity at the surface.
given that a proton is equal in weight to hydrogen.
No. A hydrogen atom weighs the same as a proton (more or less), but that’s like saying a cup of styrofoam beads weighs the same as a cup full of sand, because each grain of sand weighs the same as each styrofoam bead.
Protons sans electron would be a lot more dense than a hydrogen atom, thus for the same volume, the weight would be much much greater.
I think the neutron star referenced above is a close answer.
Which of course adds a twist. If I only put 2 protons in a “cup” then the weight will be small. If I close pack them then it’s quite large. We may need a clarification. 
And this is why it’s preferable to use weights of dry ingredients in baking rather than volume measures.
Wait, what?
What kind of cup could contain them?
But protons wouldn’t stay together like neutrons do. Protons are electrically charged, and repel each other. You need neutrons to hold any large collection of protons together. A cup of protons packed to the density of neutronium would explode.
The Neutron star answer requires the additional presence of the mass of the rest of the neutron star to generate the gravity well necessary to create the density.
Two Protons, not of the same atom will be the same distance from each other whether or not they have electrons in “orbit”.
For some general numbers, one cup would be 1/16 of a gallon, or approximately 14.4 cubic inches.
One cubic inch is approximately 16.4 ccs.
So. . . about 236 ccs.
YMMV.
Suffice it to say, though, it would be pretty heavy, realizing that this is just a thought experiment.
True but if you’re shooting for a close enough answer for close packed protons in a cup (New from Campbells!) then it works just fine. Otherwise the whole family of answers (a teaspoon of neutron star material weighs as much as a mountain) winks out of existence in millions of children science books.
At Neutron Star Density, a cup of any element would weigh the same.
Mostly because they’re not elements at that density.
Setting aside issues such as charge and significant figures:
The volume of a proton is 1.5x10[sup]-41[/sup] m[sup]3[/sup]
1 cup is 0.000236588237 m[sup]3[/sup]
Thus, 1.577254913x10[sup]37[/sup] protons will fit in one cup.
Each proton weighs 1.676621637x10[sup]-27[/sup] kg, so the total mass of all those protons would be 26,381,506,951 kg. Since there are 2.2 lbs / kg, then, 1 cup of protons would weigh about 58 billion pounds.
How much does the recipe call for?
One made of solid hypotherium.
Maybe one of our resident brainiacs will check in here. If I hold a proton in each hand, the predominant force is the repulsive force of the electric charges, right? As I bring them closer and closer, does an attractive force (weak or strong?) overcome it?