When you think of space missions, you might think of humans walking on the moon or rovers rolling around on Mars. But there’s a whole bunch of other awesome space missions, too. One of these events was the NASA OSIRIS-REx spacecraft touching an asteroid. It didn’t just touch the asteroid named Bennu, it also picked up some material from the surface. It will then return that material to Earth so that humans can study it.

Now for the cool physics. How does OSIRIS-REx determine how much material it collected? After it uses a robotic arm to “poke” the asteroid, it backs up and then spins. Yup. By looking at the change in rotational rate for the spacecraft, you can determine the amount of extra mass in the collector arm. Let me go over all the important pieces to this physics puzzle.

Moment of Inertia

The key to this whole thing is the moment of inertia. But to really understand the moment of inertia you need to look at mass. You can think of mass as the amount of “stuff” in an object—but it also has to do with the change in motion (inertial mass). Let’s try a simple experiment. Grab a full bottle of water in one hand and an empty water bottle in the other. Now shake them both back and forth.

Surely you can feel a difference. It’s not about motion, it’s about CHANGE in motion. Consider the time when a full bottle of water is moving to the left. In order to move it back to the right, it has to first slow down and stop, then speed up in the opposite direction. This change in velocity is an acceleration where the acceleration (in one dimension) is defined as:

Both water bottles accelerate—but clearly there is a difference. The full bottle of water (with a greater mass) requires a larger force to accelerate. Here you can see the connection between force, mass, and acceleration (often called Newton’s Second Law). This is that equation in just one dimension (but it’s really a vector equation).

So, in a sense the mass is an object’s resistance to changes in motion. The greater the mass, the harder it is to change its motion. But what if we rotate an object instead of shaking it back and forth? It’s time for another experiment. This one is slightly more complicated to set up since you will need four full water bottles and two sticks of equal length (I’m going to use some PVC pipes). For one of the sticks, take two water bottles and tape them to the ends of the stick. For the other stick, attach the bottles near the middle of the stick. Now you have two sticks—each with two water bottles. The mass of both water-bottle-stick combos are the same.

Now hold each stick in the middle and rotate them back and forth.

You can feel a difference, right? Which one is easier to wave back and forth? If you try this, you will find that the stick with the bottles near the center is MUCH easier to rotate. This is because it has a lower moment of inertia. Just like the mass of an object tells you how difficult it would be to accelerate, the moment of inertia tells you how difficult it is to change its rotational motion (technically the angular acceleration). Honestly, I think the name “moment of inertia” doesn’t really help people understand this concept. Perhaps a better name would be “rotational mass” since it is just like normal mass except for rotational motion.

It’s important to realize that the two sticks have the same mass, but different moments of inertia. The moment of inertia depends not just on the total mass, but where this mass is located with respect to the axis of rotation. Imagine the you have a bunch of individual masses (rather than a continuous distribution like a stick). If each mass has a distance *r* from the axis of rotation then the moment of inertia can be calculated as the sum of the products of mass multiplied by distance squared. That sounds terrible, so I will write it as the following equation instead. Yes, we use the symbol *I* for the moment of inertia.

I know that looks bad—but it’s not too terrible. The Σ just means sum all the parts (labeled with a changing index *i*), plus it looks cool. So with this definition, you can see why the two sticks have different moments of inertia (when rotated about the center). For the stick with the bottles at the end, the distance values (r) are greater and when you square them they get even bigger. That means the outer bottle stick has a larger moment of inertia which makes it harder to change its rotational motion from one direction to the other direction.

OK, just a quick comment. The above expression for the moment of inertia assumes there is a fixed axis of rotation. It’s possible that an object can rotate in very complicated ways and that expression just won’t work—but it’s still fine in this case. But now that we know something about the moment of inertia, how do we use it to measure the mass of some asteroid dirt?

There are two more things to think about: torque and angular acceleration. Let’s start with torque. In short, it’s the rotational equivalent of a force. Just like a force causes changes in linear motion, a torque causes a change in rotational motion. If you push on an object (off center), you will exert a torque. That torque depends on magnitude of the force and the torque arm. The torque arm is the perpendicular distance from the point of rotation to the location where the force is applied.

Suppose you have a spacecraft (maybe like OSIRIS-REx) and you have a rocket thruster that exerts a force. Maybe it looks like this.

Yes, we use the Greek letter τ for torque—because it’s fun. Now we can put this together to get the rotational version of Newton’s second law. Instead of net force equal to the product of mass and acceleration, we get torque (rotational force) is equal to the moment of inertia (rotational mass) multiplied by the angular acceleration.

In this expression, α is the rate of change of the angular velocity (ω).

It’s just like the linear acceleration except that it deals with angular quantities. OK, but how would all of this stuff work together to find the mass of the material from the asteroid? For this, I’m going to set up a simple experiment. It’s going to be just like OSIRIS-REx, but with simpler tools.

Here is the basic idea. I’m going to start with my spacecraft—it’s going to be this low friction rotational platform (sort of like an office chair without the seat). The platform will have an arm sticking out just like the one on OSIRIS-REx. On this platform, I’m going to mount a fan. The fan will exert a constant strength force and since it’s at some set distance from the rotation point, it will also exert a constant torque. This constant torque will cause an angular acceleration so that I can find the moment of inertia of the platform. Next, I’m going to repeat the whole thing with an extra (tiny) mass on the end of the collector arm and figure out the mass based on the new angular acceleration.

OK, the first thing I need to do is to determine the torque exerted by the fan. I could just measure the force from the fan and the distance from the point of rotation and then calculate the torque. However, since the fan has a finite size it’s not completely obvious that it has a single location for the applied force. Instead, I’m going to turn the fan on and connect a force probe to the edge of the platform so that it can’t rotate. In this case the torque from the force probe must be equal to the fan torque—but there is a big difference, the force probe has contact at just one place on the platform so that I can measure its torque arm.

Here is a picture of the setup.

From this, I get a force of 0.099 Newtons with a torque arm of 22.2 cm. This means the torque from the fan would be equal to 0.022 N*m. Now I can let the fan push the platform to change its rotational motion. But wait! You might think it would be great to just turn on the fan and let it go. No, there’s a better way. I’m going to turn on the fan (which pushes in a clockwise direction) and then use my hand to spin the platform in a counterclockwise direction. This means the fan will first slow down the rotation and stop it before rotating in the opposite direction. The nice thing about this is that since the platform rotates both ways, we can easily see if there is a significant torque due to friction as it would make the two angular accelerations slightly different.

Oh, how do you find the angular acceleration? I’m going to use my phone to get a top-down video. With that I can use a marked spot on the platform to measure the angular position in each frame (I’m going to use Tracker Video Analysis—it’s free and awesome). With a plot of angular position as a function of time, we can use the angular kinematic equation for an object with a constant angular acceleration.

This says that an angle-time plot should be a parabolic function. If I fit a quadratic equation to the data, the coefficient in front of the t^{2} term should be 1/2 times the angular acceleration (α). This is what that looks like for the first run (without an extra mass on the grabber arm).

From the fit, I get an angular acceleration of 2*0.1097 = 0.2194 rad/s^{2}. Now that I know the angular acceleration and the torque, I can find the moment of inertia for the rotating platform (including the fan itself and the grabber arm).

Notice that I added a “1” subscript to both the angular acceleration and the moment of inertia. When I add the extra mass, I’ll get a new values with a subscript of “2”. Here is the plot of the angular position of the platform with the extra mass.

Notice that I now have a slightly smaller angular acceleration with a value of 2*0.1064 = 0.2128 rad/s^{2}. With the same torque as before, this gives a new moment of inertia of:

Notice that it’s just slightly higher. It’s a higher moment of inertia because of that extra mass on the end of the grabber arm. If I consider this extra mass just to be concentrated into a single point (a fair approximation), then the new momentum of inertia can be found from the old moment of inertia with this added part.

In this expression, *m* is the extra mass and *r* is the distance of this mass from the rotation point (I measured this to be 0.222 meters). Now I know everything except for *m*—which I can solve for.

That’s a mass of about 60 grams—you know, about the mass of a size-C battery. OK, that’s could be considered a reasonably small mass in some cases but it’s quite a bit larger than it should be. Here is the item I put on the end of the grabber.

Yup. It’s a penny. With the tape, it has a mass of about 3.1 grams. I’m off. But it’s OK. My result might be off—but the method is legit. This is essentially how OSIRIS-REx determines the amount of material captured from the asteroid Bennu, but only better. The space craft has a more controlled method for finding the torque and a better way to measure the angular acceleration (probably using a gyroscope). But at least maybe now you have a better understanding of the moment of inertia.

*Formulas and imagery by Rhett Allain.*

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