How to measure mass using centripetal force? I was just reading through Volume I, Chapter $9$ of the Feynman Lectures, where he's discussing Newton's dynamics. He says,

We use the term mass as a quantitative measure of inertia, and we may measure mass, for example, by swinging an object in a circle at a certain speed and measuring how much force we need to keep it in the circle.

What is an experimental setup that could accomplish this?
 A: Suppose you have a way to measure force, a simple method would be having a spring with known proprieties.
Now tie the mass on one end of the spring and fix the other end.
When the mass is rotating with constant speed and radius, you can get the expression
by simply equating 
$$k(r - L_0) = \frac{m v^2}{r}$$
and solving for mass.

That will permit you to calculate the mass knowing: $v$: the linear speed of rotation, $r$ the radius of rotation, $L_0$: the length of the spring when there is no force on it.
A: You could very well use an elastic band tied to the object whose mass you want to measure. Or even more precisely some inextensible rope/wire and a spring with a graduated guide which allows you to read the extension of the spring (maybe using a comoving camera to take the reading).
In the most practical terms and with few pretenses on the precision you could use only a set of identical elastic loops, a 2l bottle of water, a meter and a friend with a cronometer and good reflexes.
Sketch of the experiment:
-first: determine the maximum tension of the elastic loops.
You tie the filled bottle of water to an elastic band and try to lift: if the elastic tears up you repeat (you can use the same elastic if it is not too strained) with some less water (you can extimate the weight by measuring he volume filled with the meter. Until you find the limiting value. If the elastic can lift the filled bottle you can either add another bottle (maybe not filled) or you can extimate the maximum tension by breaking the elastic with a free fall of the bottle, measuring the height that the bottle falls from (this is feasible but a bit more complex..). At this point you can also measure the maximum length $L_{max}$ of the elastic band before it tears (you can use another elastic loop for this).
-second step: you tie your object of unknown weight to the elastic and start slowly swinging the object in circles (caution to move the least possible your hand). Your friend starts measuring the period of the circular motion and as he does that you start slowly speeding up.
In this way if you measure the period of the circular motion right before the elastic band tears up and the object hots your friend, you can derive the mass of the object: indeed you know that the tension reahed the maximum value you measured and you know that the tension will be $\tau=m (g+L_{max}\omega^2)$, where $\omega$ will be the angular velocity measured bu your friend (2pi/period).
Of course you could also use a pendulum like oscillation if the object if heavy enough. The experiment does not change fundamentally.
Note that even if I suggest using elastic loops you should always cut them before using them and not using them doubled.
Also I suggest collecting a statistically relevant ebsemble of measurements to fight the miriad of experimental weaknesses of this setting..
Have fun!!!
A: Maybe something like this: a digital force gauge connected to a shaft, with a thin but sturdy metal bar of known length $r$ attached to the gauge. Attach the object whose mass you want to measure to the other end of the bar. Spin the shaft, gauge, bar, and the object at a fixed angular speed $\omega$ and record the force reading $F$ from the gauge. The next step would be to extract the mass of the object from the data. In the very simplest model (too simple), you would only need $F$, $r$, and $\omega$. But to do this for real you'll have to account for gauge and the rod since their moment of inertias will contribute to $F$.
