A very thin hoop is placed at the top of a ramp. Will the hoop (start to) roll without slipping?
Without doing the experiment, what would you want to be able to measure to help you decide? Why would these measurements help? How would you go about measuring them?. Once you’ve measured these things, how will you use those measurements to help you decide.
Teach. Brian. Teach. 
Yes it will roll without slipping.
The centre of mass is “ahead of” the point of contact between the hoop and the ramp.
The resulting torque causes its angular velocity to increase.
I think your argument implies there will an instantaneous change to angular velocity about the point of contact. I’d argue this could be achieved with sliding alone, no rolling necessary.
So, naturally, any sliding round object will attempt to, after a while, roll without slipping, since rolling without slipping means that its point of contact isn’t moving, and so it won’t be suffering any losses due to friction.
But, you seem to be interested in whether the hoop with start to roll or slide. For that we need to decide whether the friction force at the ramp can provide enough force to keep the hoop from sliding at the beginning of the motion.
Ff = mu Fn = mu m g cos theta,
where theta is the angle of the ramp
This is the maximal frictional force the ramp can provide, which needs to balance the parallel component of the gravitational force to prevent the hoop from starting to slide, that force is
Fpar = m g sin theta
So our condition for the hoop to start off sliding rather than rolling would seem to be
sin theta > mu cos theta
which is nicely independent of the mass or size of the hoop.
for a mu of between 0.25 and 0.5 or so, this tells us that if the ramp is more angled than
theta ~ 15 or 30 degrees respectively, the hoop will initially slide rather than roll.
This seems to agree with intuition.
It’s a nice argument, and I think I follow your line of reasoning. However, I think your argument about net forces being zero implies there would be no acceleration of center of mass down the ramp, right? At least initially. If there is no acceleration of center mass, then either it doesn’t move at all, or the center mass doesn’t move but somehow still spinning. The first case, it’s certainly not rolling. The second case doesn’t seem to be rolling either, since it implies slipping– the hoop would be spinning without translating, causing point of contact to slip…
I’m gonna go all meta on you here.
As a layperson w/r/t physics, I’m not sure I get all of the assumptions built into the question, and I definitely don’t get which ones I’m supposed to pay attention to or ignore.
From my lay perspective, all hoops roll on ramps. Why wouldn’t the very thin one? (And by the way, what does it mean to be very thin, anyway? Thin like a sheet of paper? Or thin AND narrow?)
All of which serves to remind me of how important it is in my own teaching to make sure that my novices and (relative) experts all have some entry point into a mathematics task in my classroom.
Thanks for the think.
Yeah, Chris, this is certainly not a good question in that sense… I like it for the thinking it made me do. It’s certainly not inviting.
What I like about this kind of question is the ambiguity. It generates a whole slew of new questions that makes one revisit many physics concepts. Great for generating discussion.
Note: my suggested solution may very well be incorrect but it would be one of several put forward by my students and then… the defense begins. The level of engagement is high because they are invested in their position and rich discussion ensues. Also serves as a great revealer of misconceptions.
Totally agreed. The arguments are what are interesting, and the plausibility of many competing arguments makes it even better. I’m not fully convinced of my best argument… which is net force (driving the center of mass acceleration) and net torque about the center of mass (driving the angular acceleration about the center of mass) have to be in lock step, ensuring the angular speed and translational speed stay at rolling-w/o-slipping condition. For the hoop, this makes it so our answers about tan(theta) are off by a factor of two, because you insisted that the net force must be zero, where as I didn’t… Anyway, there are also lots of interesting questions that arise. I spent a fair amount of time exploring “What’s the relationship between radius of gyration, angle of ramp, and coeffecient of friction for starting to roll with out slipping?”