Re: How to build a yoyo with a maximum return height

Hello,

That sounds like a fun project! I've never tried to optimize a yo-yo, so I don't quite know the best way to do it. You should certainly experiment with a number of designs, and try to figure out for yourself which aspects are important.

I can give you some general physicsy advice, though. A yo-yo is a device that converts potential energy into kinetic energy and back again. When it starts 10 meters off the ground, it has a certain amount of "gravitational potential energy" (E_grav = m g h. Energy equals mass times acceleration-of-gravity times height). When it's at the bottom of the string and spinning very fast, it has kinetic energy, specifically "rotational kinetic energy" (E_rot = 1/2 I w^2. Energy equals one-half of the "moment of intertia" times the angular speed squared). As the yoyo climbs back up the rope, some of the rotational energy is turned back into gravitational energy.

If the yo-yo were an ideal, lossless, frictionless system in a vacuum, it would climb all the way back up 10 meters every time: if energy is conserved, ALL of the original gravitational energy becomes rotational energy, and ALL of the rotational energy would go back to gravitational energy. In real life, though, you've got friction against the string, drag against the air, and some energy lost in a "jolt" when the yoyo turns around. Your job as a yoyo designer is to minimize the importance of drag; to do this, you'll want your yoyo to be very smooth. Also, if you can make your yoyo as heavy (and as high a moment-of-interia) as possible, then it will have a large amount of initial potential energy, a large amound of rotational kinetic energy without needing to spin too fast, and the energy lost to friction (which depends on the size and speed, NOT on the mass) will be a smaller FRACTION of the total energy of the system. But it sounds like you've already thought of that, and you want to make your yoyo very massive.

OK, let's think about the jolt when the string runs out. An accurate calculation is probably very difficult! You're better off figuring it out experimentally! But let's see if we can do anything on the back of an envelope:

Hmm. You can figure out the yoyo's final angular velocity, "w", using conservation of energy and the equations above. (Or you can measure it). Let's just figure out how fast the yo-yo changes directions at the bottom of the string. In the yo-yo's last quarter-rotation, the center-of-mass is moving downwards, and the string is (say) just to the left of the axle And we'll write the height z and the time t in terms of the radius of the axle R and the angular velocity W. (W is in radians per second; the number of revolutions per second is 2*pi*w)

     |
     |    <-- string   t = -Pi/ (2 w)
     |                 z = +r
      O   <-- axle

     |                 t = 0
     |                 z = 0
     |
     O

     |                 t = +Pi/(2 w)
     |                 z = +r
     |
    O

That's a real back-of-the envelope calculation of the "average force during the last 1/2 rotation". There's also the force due to gravity, which is constant, that you can just add to the above. I hope you can go through this and get all the units right, and get an understandable number to try out on your yoyo. In terms of minimizing the jolt in your yoyo design: notice that the force is proportional to "r", the radius of your axle. A smaller axle will give a smaller jolt.

Let me end with a disclaimer. I hope you can see how many assumptions went into this calculation: I ignore the stretching of the string. I wrote down a quadratic equation for the height of the yoyo during that last rotation - is that justified? I don't calculate whether or not the yoyo actually falls during that last quarter turn; in principle, if it is spinning fast enough, the string might come over the top of the axle and start pulling the yoyo up _before_ it has fallen by a distance r. So, there are all sorts of things that might generate stronger or weaker momentary forces in a yo-yo string! A "simple physics" calculation is fun, it is a good place to start thinking, but don't trust your life (or your success in a school project) to it! Check my back-of-the-envelope answer against your intuition, and against a real experiment!

Good luck,

-Ben