Rotating Frames of Reference
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Rotating Frames of Reference

October 14, 2019

Here you see footage of hurricanes that formed
in the Atlantic Ocean during 2009. Do you see that every single hurricane rotates counterclockwise?
All hurricanes formed in the Northern hemisphere rotate counterclockwise. In this video, we’ll provide you with the
tools to explain why hurricanes rotate the way that they do. This video is part of the Representations
video series. Information can be represented in words, through mathematical symbols, graphically,
or in 3-D models. Representations are used to develop a deeper and more flexible understanding
of objects, systems, and processes. Hi, my name is Sanjay Sarma, and I am a professor
of Mechanical Engineering at MIT. Today we are going to demystify the origins of the
forces that appear to act on objects in rotating frames. Before watching this video, you should be
familiar with how to define basis vectors; inertial and non-inertial reference frames;
and the representation of rotation rates as a vector cross products. After watching this video, you will be able
to explain why centrifugal and Coriolis forces arise in rotating frames of reference, and
apply your understanding of the Coriolis force to determine the direction of rotation of
hurricanes. A frame of reference is a choice of coordinate
frame, a set of orthonormal basis vectors. The frame is allowed to undergo rigid body
motions. Rigid body motions include translation, rotation,
or a combination of translation and rotation. As a frame of reference undergoes a rigid
body motion, the 3 basis vectors retain their unit length and remain mutually orthogonal. In this video, we are going to focus on rotating
frames of reference. In particular, we want to think about frames that are rotating with
constant angular velocity and that aren’t translating. Rotating frames of reference are non-inertial,
thus we detect fictitious forces in them. We are going to explain how these so called
fictitious forces arise. This is a turntable. We say that we are in
the turntable frame of reference because the camera is mounted to the turntable. From this
frame of reference, it appears that the turntable is stationary as the world spins around us,
despite the fact that our experience tells us it is the turntable that is spinning. Here you see an orange disk attached to the
turntable by a string. If we rotate the turntable quickly enough to overcome the friction between
the disk and the turntable, you notice that the string becomes taut. Newton’s second law implies that there must
be a force equal and opposite to this tension force for the disk to remain stationary! This apparent force is what we call a fictitious
force. Let’s rotate the turntable with two disks—one is attached to the table by a
string, while the other is unattached. The fictitious force causes the unattached disk
to fly off of the Turntable. Is there really a force? If so, where does
it come from? Pause the video and discuss. In addition to the Turntable frame, there
is another frame of reference that will be useful in our analysis. This is the ground frame, G, which is any
coordinate frame that appears to be stationary while standing on the ground. The Turntable frame, T, refers to any coordinate
frame that appears stationary while standing on the Turntable. From the ground frame, the T frame is rotating
counterclockwise with constant angular velocity, or almost constant angular velocity. It is important to be explicit about the particular
frame of reference used to describe a velocity or acceleration vector. The velocity of the stationary vector is zero
in the G frame, but the velocity of the rotating vector is zero in the T frame. We will denote the frame of such a vector
by using a left superscript G or T to designate if we are considering the vector as an object
in the G frame or the T frame. In the T frame, the position of the disk is
fixed, so its velocity and acceleration are both zero. Pause the video here to determine the velocity
of the disk as seen from the ground frame. According to the G frame, the T frame is rotating
with some angular velocity, represented by the vector omega. Omega points along the axis
of rotation with magnitude equal to the angular velocity. The velocity of the disk according
to the G frame is entirely due to the rotation of the T frame, and thus can be represented
by omega cross r. Recall that in general, any object that is
moving on the turntable will have a velocity that can be defined in the T-frame. The velocity
in the G-frame can be found as the sum of the velocity in the T-frame and the velocity
that arises due to the rotation of the T-frame with respect to the G-frame. We can think of this as a rule for taking
the time derivative of the position vector r in both the G-frame and the T-frame. We can generalize this as a formula for how
to take the time derivative in the G frame of any vector, x, in terms of its T-frame
derivative. This formula holds when the T frame is rotating, but not translating with
respect to the G frame. Use this formula to take the time derivative
of velocity. Try to find the general formula for the acceleration in the ground frame in
terms of the acceleration in the turntable frame and various other terms. Pause the video
while you carry out the computation. Looking at this formula, the first term is
the object’s acceleration as observed in the T frame. The remaining terms can be thought
of as giving rise to the fictitious forces, which cause the acceleration observed in the
T frame. The second term is the due to the angular acceleration of the T frame with respect
to the G frame. This third term is the Coriolis acceleration—note that it depends on the
velocity vector of the object in the T frame. And this last term is the centripetal acceleration—observe
that it depends on how far the object is from the axis of rotation. Given this information, pause the video and
determine the acceleration of the orange disk in the ground frame? The velocity and acceleration of the disk
are zero in the T frame of reference. We did our best to rotate the turntable in
this video with constant angular velocity. So we will assume that the angular acceleration
is negligible. Thus the acceleration in the G-frame is given
completely by the centripetal acceleration term. This vector points in the negative r direction
with a magnitude given by the distance from the axis multiplied by the angular velocity
squared. In the G frame, which is inertial, we do not
observe effects of the so-called “fictitious forces”. Remember, the disk wants to move
in a straight line. It doesn’t want to turn. The string imparts a tension force upon the
orange disk, which provides the centripetal acceleration needed for the disk to rotate
with the Turntable. But in the non-inertial T frame, we may think
we observe a “fictitious force”. The only thing that is fictitious is your perception
that the disk is not accelerating. It is a “physical illusion” created by the fact that
from the turntable frame of reference, you don’t observe the centripetal acceleration
of the turntable. In this video clip, we are rolling tennis
balls through a plastic tube. In the T frame, we observe that the ball moves along a curved
path. The curved motion observed in a rotating frame of reference is called the “Coriolis
Effect.” Pause the video and explain what you think
is causing this curved motion. To better understand what is happening, let’s
look at the same motion from the inertial ground frame. But first, what do you think the motion of
the ball will be when observed from the Ground frame? Pause the video and make a prediction. That’s right. The path is a rather straight
line along the initial trajectory! It is the motion of the turntable that is
curved as the turntable rotates counterclockwise. But from the turntable frame, points on the
turntable appear stationary, because you are rotating with the frame. So you perceive that
the ball is curving to the right, even though it is you that is moving in a circular path. You might try to explain the curvature using
fictitious forces. But we know that these forces are really just accelerations of the
turntable frame that we do not perceive. Let’s use what we know about the G frame to
understand what these accelerations are in the T frame. Using the general formula for acceleration
in a rotating frame that we found earlier, pause here and determine the acceleration
in the T frame. Because the velocity is essentially constant
in the G frame, the G frame acceleration is zero. We can rearrange the remaining nonzero terms
to find an expression for the acceleration of the ball that we observe in the T frame.
We find it is equal and opposite the sum of the Coriolis acceleration and the Centripetal
acceleration. This term is the negative of the Centripetal
acceleration, which we saw in the previous example. This acceleration is always pointing
outwards from the center of rotation. This second term is the negative Coriolis
acceleration. This acceleration is perpendicular to the velocity of the ball in the T frame,
creating the curvature of the ball’s path. We’ve said the accelerations are created by
the rotation of the turntable. How does this happen? For an object to rotate with the turntable,
a centripetal acceleration is required. Without it, an object appears to move outwards from
the center of rotation. Even more is happening though. Because of
the rigid body rotation, the velocity of a point near the outside edge of the turntable
is greater than the velocity of point near the center of rotation. Moving objects accelerate due to this velocity
differential. The acceleration is perpendicular to the velocity of the object. But you don’t
realize this acceleration exists from the turntable frame, because you don’t perceive
your own rotation. The acceleration created by this velocity
differential between points on the turntable is responsible for the Coriolis effect, curving
the paths of moving objects. You can feel these accelerations yourself
if you walk around on a carousel. Or you can try waving your arms or legs on some other
rotating theme park ride. Remember, the Earth is a rotating reference
frame. Even though we are used to considering the Earth as a fixed frame, some phenomena,
such as hurricanes, are created by the rotation of the Earth. We are going to use what we
learned earlier to understand why hurricanes rotate the way they do. We can model regions of the Earth quite easily.
For points that are in the Northern or Southern hemisphere, but sufficiently far from the
equator, we can model the hemisphere by projecting it onto a disk. From this perspective, the northern hemisphere
is a counterclockwise rotating disk, and the southern hemisphere is a clockwise rotating
disk. Near the equator, we can’t model the Earth
as a disk. Instead, a better model would be the outside surface of a rotating cylinder. Our understanding of the Coriolis Effect from
the turntable will directly apply to this model of the hemispheres. Now let’s start by thinking about hurricanes.
Hurricanes are formed when there are small regions of very low air pressure. The existence
of this low-pressure region causes air from all directions to move towards the low-pressure
zone. With this information and what you know about
the Coriolis effect, explain why hurricanes in the northern hemisphere rotate counterclockwise. As air moves towards the low-pressure zone,
the air moving from the south veers to the right due to the Coriolis effect. Similarly,
air moving from the north veers to the left of the low-pressure zone. At the same time,
air is constantly pulled in towards the low-pressure zone. The combination of these processes creates
a region of counterclockwise rotating air. This is why hurricanes in the Northern hemisphere
rotate counterclockwise. In this video, we saw that in rotating frames,
an apparent force pulling away from the axis of rotation is really the objects tendency
to move in a straight line. In a rotating frame of reference, objects
require a centripetal acceleration to remain stationary in that frame. Moving objects in rotating frames move in
curved paths due to the Coriolis acceleration, created by the fact that the velocity of points
in the frame are greater further from the axis of rotation. On Earth, we can observe the Coriolis Effect
in the counterclockwise rotation of hurricanes in the northern hemispheres. This leaves 2 questions for you. What direction
do hurricanes rotate in the southern hemisphere? And why can’t hurricanes form at the equator?

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