Balancing a yardstick

Question

There is a known trick to this, so “no computers” may be used to look up the answer.

Alice gives Bob a wooden yardstick with a small weight secured to one end and says, “See if you can find the center of balance of this yardstick, so it rests horizontally on your finger at that spot.” Bob tries to balance the yardstick with his finger at the 18-inch mark, but the weighted end drops quickly. Through trial and error, Bob finds that it balances around the 14-inch mark. Alice says “Not bad, but it took you a while. Here’s another yardstick with a different weight. See if you can find the center of balance in one motion, without guessing, by touching the yardstick only with your hands.”

How is it done?

Answer 1

I'll refrain from giving the answer I know from before, since I didn't originally figure it out for myself, so giving that answer would feel a bit like cheating. Instead, here's how you might find the balance point using one hand only:

grab the stick at one end, and toss it in the air, giving it a hefty spin, like a juggler's club or a majorette's baton.

Now you can actually see the centre of balance:

while no significant forces affect the spinning stick, all rotation will happen around its centre of mass. (If the rotation happened around some other point, the centre of mass would be constantly moving in circles, violating Newton's first law.) This means that the spot that isn't blurry or whirring about is the centre of mass.

(If you are very agile, you may even be able to catch the stick by grabbing that spot)

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EDIT: since it's already been a week and no-one seems to be willing to give the well-known answer, I'm adding it below. I didn't invent it myself, but have played with the effect quite a lot, it kind of looks like a magic trick when done smoothly.

You can find the centre of mass by utilising

friction and gravity, with some help from lever arms.

First, you grab the yardstick by both ends, and

support it horizontally on top of your hands. (For consistent results, you'll want to use the same part of the same finger in both hands to support the stick near its ends.)

Then you slowly but steadily

bring your hands together

and wonder of wonders, the stick will

turn into a self-centring automaton. Here's why:

Since the two supporting hands are pushing the stick in opposite directions, the stick will slide over the hand on the side that has less friction. Assuming similar coefficients of friction and a similar area of contact, that will be the side where the force of the stick pressing down on the hand is smaller.

However, when the stick slides over the hand like this, the force acting on that hand must increase: the net torque must remain zero (the stick is not spinning), and the lever arm (relative to the centre of mass) is getting shorter.

When the supporting force increases (and correspondingly decreases at the other hand: the stick is not moving vertically), the amount of friction will also increase, and at some point, it will be the other hand that starts to slide, and the stick will change direction.

This means that after some going back-and-forth, the stick is very likely to end up

still balanced on your hands when they come together

which means that

the centre of balance will be the spot that's right over your hands.