Atwood Machine: what if the rope is massive?

The question was asked on a physics teachers message board: How do you deal with a rope-and-pulley situation if the rope isn't truly massless?

I'm giving the AP Physics 1 answer here today.  This video by Pasco's Dan Burns gives a seriously intense mathematical solution.  You don't need that unless you're in and beyond AP Physics C, or unless you're solving the 2022 US Invitational Young Physicists Tournament problem on "rope and chain fountains."

I'm thinking of a modified atwood - one cart on a horizontal low-friction track, connected by a rope over a pulley to a hanging object.

Consider the entire cart-rope-hanging object system.  The unbalanced external force on the system is the weight of the hanging object.  (The force of the ropes on the objects is internal, because the rope is part of the system.)  Then set the system acceleration equal to this unbalanced force divided by the system mass.

But now give the rope a significant mass, still smaller than the mass of either object.  The unbalanced external force doesn't change (much)* - still the weight of the hanging object!  But with a larger system mass in the denominator, the acceleration becomes smaller.

*Okay, okay, the weight of the rope that's dangling contributes to this unbalanced external force.  So start the system with most of the massive rope horizontal.  And the rope's mass is still small compared to the hanging object's, so this is a good approximation.

Test this in class!  Use a smartcart on a track, connected by a thread over a pulley and onto, say, a 50 g hanging object.  Use the smartcart to measure acceleration.  Then remove the thread, and instead use a chain - like a bike chain, maybe? - to connect the smartcart to the hanger.  I'll bet you get a smaller acceleration!

This AP live video from April 2020 shows me doing this experiment with a low-mass rope, if you need ideas of how to make the acceleration measurement experimentally.