# How can a yo-yo de-spin maneuver reverse the rotation?

I just read about yo-yo de-spin as a measure to reduce the spin of objects. The basic idea is simple and makes sense, but then I read this:

As an example of yo-yo de-spin, on the Dawn Mission […] reduced the initial spin rate of the 1420 kg spacecraft from 36 RPM down to 3 RPM in the opposite direction.

The linked source also mentions this:

Even with a 204-kilogram ([…]) third stage […] and a 1218-kilogram ([…]) spacecraft, the small yo-yo system halts the spin and even reverses it, leaving Dawn rotating at 3 rpm in the opposite direction from its original spin.

How does extending the cables / weights actually cause a reversal of the rotation? What is missing from the explanation?

My gut tells me that this technique should only be able to reduce spin to zero as the extension goes towards infinity. If at some point it were to reverse the spin direction, there'd have to be a point of no spin at all which would violate the conservation of momentum, no?

It would only reduce spin if the weights remained attached.

Imagine this: two ropes wrapped on a spinning spool, with weights on the ends.

As the weights are released, they spin outwards, extending the rope. The centrifugal force pulls the rope taut (most of it still wound on the spool). As they continue to spin outwards, the pull of the rope on the spool caused by the centrifugal force creates a torque on the rocket - pulling to unwind more rope faster, as result rotating the spool (and the rocket) in direction opposite to original. This torque makes the rocket spin in the opposite direction.

Now as the rope ends, normally it would start winding back on in reverse, then again unwinding, and so on, the harmonic cycle dying down due to losses in flexibility of the rope, until the whole thing ends up spinning slowly in the original direction with the weights fully extended. But when the rope ends unwinding, it's released and the weights float away, and nothing stops the reverse spin of the rocket.

In the pic below, the whole thing originally turns clockwise. As the pull on the ropes increases, the weights continue to turn clockwise, traveling outwards, but the torque from the ropes causes the hull to start spinning counter-clockwise.

• "as result rotating the spool (and the rocket) in direction opposite to original" Wouldn't that wind the rope back up on the spool? I hope it's clear that this technique does in fact reduce the spin, and the ropes don't stay attached, so I am also confused by your first sentence. The spin reduction results from the transfer of angular momentum. Jun 13, 2016 at 0:59
• @OrganicMarble: No; it's wound in such a way that to unwind it the weighs must spin at higher angular velocity than the rocket. See the picture and imagine it whole spinning clockwise. The ropes continue around the hull (not visible in the drawing). If you still don't understand, it will have to wait until tomorrow. It's 3AM here and I'm really in no condition to offer a better explanation now.
– SF.
Jun 13, 2016 at 1:24
• Maybe you should take a look at this: youtube.com/watch?v=HCtNqD-jlPE Jun 13, 2016 at 1:28
• @OrganicMarble: A nice video. Now imagine what happens if you increase the length of the string by some half the circumference's worth: the spool will start turning in the reverse direction.
– SF.
Jun 13, 2016 at 7:27
• There's actually another great video here. At 6:30 you can actually see the spin reversal process in action. Pretty cool! I especially loved how seeing it in action suddenly made it feel completely obvious, as opposed to the feeling that it was counter-intuitive I had originally. Jun 18, 2016 at 13:18

NASA - and math - to the rescue!

2nd edit: Shorter answer even "firster". Try a Gedankenexperiment. If you are happy with it stopping, then just image the satellite is suddenly much lighter. The ropes keep pulling and yo-yo's keep moving about the same, but the satellite slows down much faster. It stops and starts turning the same way the ropes are pulling it.

edit: Short answer first. Just like in linear momentum recoil, you conserve energy and angular momentum here. Whatever the objects need to do to conserve them, that's what happens. There isn't a different "why" for each new conservation problem. However, there may be an "aha!" specific to each person, after they solve enough conservation problems in a row.

Yep, at least according to Equation 3 (shown below) in NASA technical note D-1420 (1962), you can go beyond zero and reverse the spin as much as you want, provided you can manage the logistics increasingly heavy yo-yos, long wires, large yo-yo-deploy radius, or increasingly small satellite moment of inertia. NASA technical note D-1676(1963) develops the math further for stretch yo-yo techniques.

$$\frac{1+r}{1-r}=\frac{I}{m(d+a)^2}$$

$$r=\text{spin reduction ratio (final spin rate / initial spin rate),}$$

$$I=\text{spacecraft moment of inertia about the spin axis,}$$

$$m=\text{mass of weights plus 1/3 mass of wires,}$$

$$d=\text{cord length}$$

$$a=\text{spacecraft radius}$$

I like it better upside down and reversed:

$$\frac{m(d+a)^2}{I}=\frac{1-r}{1+r}$$

I plotted the equation down to a ratio of $$r=-0.2$$. Your example of +36RPM down to -3RPM would be $$r=\frac{-3}{36}\approx-0.083$$

@OrganicMarble mentions this YouTube video in his comment. It sure looks to me like the yo-yo's are getting a slight propulsive kick beyond their inertia, but haven't analyzed frame-by-frame yet.

If the original spin is clockwise: the weights will accelerate a counterclockwise spin as they begin to unwind. This will slow the clockwise spin to a stop (if the cable is long enough; and the weights have enough mass).

If the rocket has stopped spinning and the cables are still attached, and the cable has not reached 90° to tangent, then the counterclockwise acceleration will continue. Until 90° to tangent the force in the cable is always counterclockwise.

At first the momentum is given from the rocket to the weights; but there comes a point where the weights give the motion back to the rocket. If the weights are not released the counterclockwise acceleration continues; up to 90° to tangent.

All the motion can be returned to the cylinder or rocket but only linear momentum can be transferred from small to large objects; this means that all the motion (linear momentum) of the cylinder is given to the spheres.

A convenient place to release the weight is when the tether is at 90° to a tangent line to the circle of the satellite. A tangent line touches the circle at one point.

If the rotation of the satellite is stopped when the weights are released; then there is a unique length of cable that allows this to happen. If the cable is longer; then the satellite will rotate backwards before release. If the cable is shorter; then the satellite rotation will not be stopped before the tether reaches 90° to tangent.

If the cable is of infinite length then the rotation of the satellite will stop and the begin rotating backwards until the rotation of the satellite equals the distance that the weight is traveling away from the satellite.

If the rotation of the satellite is stopped when the tether is at 90° and the weights are not released: then the weights will restart the satellite and completely restore the rotation in the original direction.

When the satellite rotation is stopped the weights on the end of the tethers have all the motion. For Newtonian momentum conservation, in the Dawn Mission, the weights need to be moving about 400 m/sec. For energy conservation they need only be moving 20 m/sec.

But this 20 m/sec is not enough momentum to even cause a 3 RPM back spin. A 3 RPM back spin is 8.3 % of the original 36 RPM; and 20 m/sec is only 5% of the needed 400 m/sec. This 3 RPM back spin is an attempt to release the tether when the rotation is stopped; and the 3 RPM back spin is only a small percentage of the total motion of the weights. This means that the weights are moving much to fast for energy conservation.

The video shows that the spin is completely restored when the weights are left attached; that means that Newtonian momentum is conserved.