When I arrived at Woodberry Forest eleven years ago, I discovered several 1970s-vintage turntables in the storeroom. I turned one into a nice, quantitative, circular motion demonstration.
In the picture you see the turntable, with five identical brass masses placed on top. Before I start the demonstration, I show the class that a brass mass placed right near the center stays put when the table rotates; but the same mass placed near the outer edge flies off. The goal of the demonstration is to predict the maximum radius for which the mass will not fly off.
The friction force on the brass mass acts toward the center of the rotation, and is a centripetal force equal to mv2/r. The variable r represents the radius of the circle -- that's what we're looking for here.
Problem is, the speed v itself depends on r, because a mass near the edge covers a larger distance in the same time as a mass near the center. So we write v in terms of the period of revolution, T: v = 2πr/T. The period of revolution is measured with a stopwatch.
The friction force can be set equal to μFn. The normal force on the mass is simply its weight. The coefficient of (static) friction must be measured... but I can do that with a spring scale or a force probe. I pull a stack of brass masses with the scale. I divide the reading in the scale just as the masses start to move by the weight of the stack of masses.
(Here the class can be asked whether the coefficient of friction for a single mass will be greater, less than, or the same as that for the stack of masses. Answer: the same, because we still have brass in contact with the same surface. I only use a stack of masses to make the spring scale reading easier to obtain.)
So now we know everthing in the relevant equations necessary to solve for the radius r. We find that radius to be in the neighborhood of 4-5 cm. That's in the middle of the second mass, as measured from the center.
What does this mean experimentally? When I turn the turntable on, the outer masses should fly off, but the first one or two should stay put. And sure enough, that's what happens -- physics works.
Follow-up: Next I replace the 20 g brass masses with 10 g brass masses from the same set. Which ones should fly off now? Answer: still, the masses more than 4-5 cm from the center will fly off, the others will stay put. The coefficient of friction still hasn't changed, because we're using the same surfaces. Mass cancels out when solving in variables, so the mass doesn't matter. Sure enough, the 10 g masses fly off outside of 4-5 cm, as predicted.