Video: Gyroscopic Precession

In this video we learn what gyroscopes are, why they are useful, why they precess, and how to calculate gyroscopic precession rates.


Video Transcript

In this video, we’re going to learn about gyroscopic precession. We’ll learn what this term means. We’ll see why it is that gyroscopes precess. And we’ll learn how to work with this precession practically.

To start out, imagine that, as an astronaut inside the International Space Station, you’d like to find a steady, reliable way of knowing which direction is up, no matter how the space station is oriented. With that goal in mind, you think back to the notion of the conservation of angular momentum. Suddenly, you have an idea. What if the way to figure out which direction is up is to use a spinning object which wants to stay spinning that same way?

To dig deeper into this topic of how do we find a stable orientation point, we’ll want to know something about gyroscopic precession. When it comes to this topic though, the first thing we may be wondering is what exactly is a gyroscope. A gyroscope is a disc that’s set in motion to spin around an axis. And this disc is made free to rotate in a constant direction.

Practically, what that means is that this rotation axis is kept in a steady orientation. Often, the way that’s done is to enclose this spinning disc along its axis of rotation with a series of rings, say made of metal or some other solid material. The rings are made free to rotate at their joints around all three directions of motion. This means that, no matter where we put our gyroscope or how we move it, the disc that spins in the center will keep a stable constant orientation.

Once we have a system like this, whose axis of rotation always points in the same direction, we now have a stable orientation point for figuring out, say, which way is up and which way is down. This constancy of direction comes in handy when we’re designing, say, a navigation system for an airborne object. Many of us have experience with an everyday object that’s very similar to a gyroscope. And that is a spinning top.

When we spin a top, we set up a rotation about an axis through its center of mass. When we spin a top, we give it angular momentum. And since angular momentum is conserved, the top tries to continue spinning as it has been. But here’s where the difference between a spinning top and a gyroscope comes in.

From practical experience, we know that a spinning top doesn’t stay that way forever. Over time, a top loses its upward orientation and starts to tilt to the side. This is because tiny nudges on the top due to gravitational torques applied to the center of mass of the top along with loss of energy due to friction cause it to destabilize bit by bit.

A gyroscope, on the other hand, used as an instrument, typically has a motor that keeps the disc rotating at a steady angular speed. And it has a rotation axis which helps to correct for slight nudges off of that axis. But with our top, as the force of gravity continues to pull on the center of mass of the top and apply a torque about its base, the angle from the upright vertical to the actual axis through the top continues to grow. And eventually, as that angle gets bigger and bigger, the top completely destabilizes and falls over.

But, interestingly, before that happens, and after the top has started to destabilize, the end of the top, where we put our hands on it to give it its initial spin, begins to slowly rotate in a big wobbly circle around its original axis of rotation. It’s this kind of movement that we call precession. That’s the slow rotation of a spinning object about another axis due to torque.

On Earth, that torque is often supplied by gravity, as it is in the case of a spinning top. One of the really interesting things about precession is the rate at which it happens. If we were to track the motion of the end of the top as it moves about in these big slow circles, we would see that it actually moves at an angular speed we could call 𝜔 sub 𝑝 for the precessional angular speed. That angular speed is different from the angular speed of the top as it rotates, which we could simply call 𝜔.

When it comes to the relationship between 𝜔 sub 𝑝 and 𝜔, the speed at which the top itself rotates about its own axis, it would be amazing if they were equal to one another. But we find that that’s not actually the case. However, there is a mathematical relationship that connects these two quantities.

We’ve talked about how the gradual tilting and eventual collapse of every spinning top is due, on Earth’s surface, to torques caused by gravity. If we draw a vector that goes from the tip of the top, where it contacts the ground, to its center of mass, we could call that the distance vector and give it the label 𝑟. If we take the magnitude of that vector and call it simply 𝑟 and multiply that by the gravitational force on this spinning top, which will be equal to its mass times the acceleration due to gravity. To solve for the angular precession speed, we divide this value by the angular momentum of the top itself as it spins.

We can recall that the magnitude of angular momentum 𝐿 is equal to an object’s moment of inertia times its angular speed. With this substitution, using the moment of inertia of the top and its angular speed about its own axis 𝜔, we now have an expression for the angular precession speed of the top about its original vertical axis, as it loses stability and begins to wobble.

Looking at this equation, sometimes it’s just as interesting to see what isn’t there as what is there. This equation has nothing in it about the angle off of the vertical axis, we could call it 𝜃, of the top’s current axis of rotation. This means that, regardless of 𝜃, as long as the top is upright, it will precess at the same angular precession speed.

This means that when 𝜃 is small and the end of our top is very close to the original vertical axis, the linear speed of the end of the top will be slow. But that linear speed will increase and increase as 𝜃 grows in order to maintain this angular precession speed. Let’s take a moment now to get some practice with this idea of gyroscopic precession through an example.

A gyroscope spins with its tip on the ground, which produces negligible frictional resistance. The gyroscope has a radius of 5.0 centimeters and a mass of 0.30 kilograms and spins at 20 revolutions per second. The center of mass of the gyroscope’s disc is at a 5.0-centimeter displacement from its tip along the rotational axis of the gyroscope. What is the precessional period of the gyroscope?

We can call this precessional period capital 𝑇 sub 𝑝 and begin on our solution by drawing a diagram. The gyroscope in this example consists of a rotating disc with a mass of 0.30 kilograms and a radius, we’ve called 𝑟, 5.0 centimeters. The center of the rotating disc is a distance, also 𝑟, 5.0 centimeters above the ground. And the gyroscope rotates at an angular speed we’ve called capital Ω of 20 revolutions per second.

We know that, under these conditions, the axis that runs through the center of the spinning gyroscope and is perpendicular to it will begin to deviate from a vertical line. As the axis of the gyroscope moves off of that original vertical line, we know that this axis itself will slowly begin to rotate about the original vertical. That rotation will itself have an angular speed we can call 𝜔 sub 𝑝, the precession angular speed.

We can recall the mathematical relationship describing that angular precession speed, 𝜔 sub 𝑝. That angular rate is equal to the mass of our object times the acceleration due to gravity times the distance from the point of contact of the object with the ground to its center of mass all divided by its moment of inertia times its own angular speed, 𝜔.

As we consider this relationship for our scenario, we’re given the mass, 𝑚, of the gyroscope. And the acceleration due to gravity, 𝑔, we can treat as exactly 9.8 meters per second squared. The distance from the point of contact of our gyroscope with the ground to its center of mass is also given to us, 5.0 centimeters. So, all that remains is to solve for the object’s moment of inertia and its angular speed in radians per second.

Knowing that our gyroscope is a disc rotating about a line through its center, when we look up the moment of inertia for an object of that shape, rotating in that way, we see it’s equal to one-half the object’s mass times its radius squared. Plugging that in to our expression for 𝜔 sub 𝑝, we see that the mass value of our gyroscope cancels out, as does one factor of its radius 𝑟. So, the angular precession speed is equal to two times the acceleration due to gravity over the radius of the disc times its angular speed in radians per second.

We’re not given the angular speed of the disc in those units, but we are given its angular speed in units of revolutions per second. Knowing that one revolution about a circle is equal to two 𝜋 radians, that means we can substitute capital Ω times two 𝜋 in for lowercase 𝜔, the angular rotation rate of the disc in radians per second. This expression will let us solve for 𝜔 sub 𝑝, but what we wanna solve for is 𝑇 sub 𝑝, the period of the precession.

We know that, in general, period is equal to two 𝜋 divided by angular speed. If we apply this relationship to our scenario, we can say that 𝑇 sub 𝑝 is equal to two 𝜋 over 𝜔 sub 𝑝, or two 𝜋 over this expression we’ve arrived at. Simplifying this expression, we see it’s equal to 𝑟𝜔 times four 𝜋 squared radians per revolution all divided by two times the acceleration due to gravity. A factor of two cancels from our numerator and denominator. And we’re now ready to plug in and solve for 𝑇 sub 𝑝.

When we do plug in, we’re careful to insert our radius 𝑟 in units of meters. Before we calculate this result, let’s take a look at the units in this expression to see that they work out. First, the units of meters in our numerator and denominator cancel one another out. And when we multiply our angular speed in revolutions per second by our angular conversion from radians to revolutions, the units of revolutions drop out. And with the factors of time in seconds involved, we can see that our final units will be radian seconds, or simply seconds. That agrees with what we’d expect for units for a period 𝑇. When we do enter these numbers on our calculator, we find a result, to two significant figures, of 2.0 seconds. That’s the precessional period of this rotating gyroscope.

Let’s summarize what we’ve learned so far about gyroscopic precession. In this section, we first saw that a gyroscope is a spinning disc that’s free to rotate in a constant direction. So, no matter the position or angle of the base of the gyroscope, the axis that goes through the gyroscope’s center will always point straight up and down.

We’ve also seen that an unconstrained spinning object tends to slowly rotate about another axis due to torque and that this low rotation is called precession. And finally, we saw that this angular speed, this rate of precession, has its own mathematical relationship.

When precession is caused by torque acting on an object, thanks to the influence of the force of gravity, the angular precession speed is equal to the object’s mass times 𝑔 times the distance from the point of contact of the object with the ground to its center of mass all divided by its moment of inertia times its own angular speed 𝜔. Gyroscopic precession is related to the conservation of angular momentum. And its application is often in helping us orient ourselves in a constant direction.

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