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.
