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Here's a well-thrown disk. But have you ever
thrown one badly, so that as it spins, there's
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a wobble? This wobble rotates with a different
frequency than the disk itself spins.
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In this video, we're going to describe mathematically
the motion of all of the points on this badly
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thrown disk.
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This video is part of the Linearity Video
Series. Many complex systems are modeled or
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approximated linearly because of the mathematical
advantages.
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Hi, my name is Dan Frey, and I am a professor
of Engineering Systems and Mechanical Engineering
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at MIT. And I use rigid body kinematics myself
when designing radio--controlled aircraft.
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Before watching this video, you should be
familiar with eigenvalues and eigenvectors,
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the standard basis, e1, e2, e3 of R3, and
orthogonal matrices.
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After watching this video, you will be able
to: identify rotation matrices; decompose
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the motion of the badly thrown disk into translational
and rotational components; and write the rotational
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motion of the disk as a product of rotation
matrices.
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Our goal is to describe the motion of the
disk. The disk is a rigid body; it doesn't
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stretch, bend, or deform in any way when it
is thrown.
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In this video, we're not interested in why
the disk moves the way it does—that is,
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we're not trying to describe torques and forces
that govern the motion. We simply want to
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describe the motion mathematically. This is
a job for rigid body kinematics. We're going
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to describe the motion by decomposing it into
translational and rotational components.
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We'll start with the mathematics of rotation
matrices.
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This will allow us to build up to a description
of the wobbly disk.
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Finally, we'll complete the description of
the wobbly disk by adding in the translational
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component.
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Let's start with some linear algebra.
A rotation is a mapping that takes any vector
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in R3 to some other vector in R3 via rotation
about some axis by some angle.
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Rotations don't change the length of a vector.
So, if you scale a vector and then rotate
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it, you get the same thing as if you first
rotate it, and then scale the vector.
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Also, if you take two vectors, sum them and
then rotate the sum, this is equal to vector
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you would get if you first rotate both vectors
and then add them.
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These two properties together mean that Rotations
act linearly on vectors.
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And by definition, linear operations can be
represented by matrices.
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But what does a rotation matrix look like?
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We can learn a lot about a matrix by examining
its eigenvalues and eigenvectors. Recall that
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a vector v is an eigenvector of a matrix if
it is sent to a scalar multiple of itself
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when acted upon by the matrix.
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That scalar is the eigenvalue.
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Consider a rotation of 60 degrees about the
axis defined by the vector e1+e2. Pause the
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video here and determine one eigenvalue and
eigenvector. By the definition of an eigenvector,
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the vector e1+e2, which points along the axis
of rotation, is an eigenvector with eigenvalue
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one. This is because this vector is UNCHANGED
by the rotation matrix.
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Suppose you have a rotation matrix such that
e1 and e2 are both eigenvectors with eigenvalue
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1.
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What would this mean about the rotation? Pause
the video and think about this. The entire
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xy-plane will be unchanged by this rotation.
This is only possible if the matrix is the
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identity matrix! This is the null rotation...
nothing happens! What are the eigenvalues
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and eigenvectors of a 180-degree rotation
about the z-axis?
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This rotation matrix has one eigenvalue of
1, corresponding to the vector e3, which points
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along the axis of rotation.
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But it has more eigenvectors: any vector in
the xy-plane is sent to its negative by the
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rotation, so any vector in the xy-plane is
an eigenvector with eigenvalue -1.
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Now let's consider a rotation by some angle
theta (that is not an integer multiple of
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pi) clockwise about the z-axis.
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Write a matrix that represents such a rotation.
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Compute the eigenvalues of this matrix, and
use the definition of an eigenvector to explain
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why this makes sense.
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You should have found 1 real eigenvalue equal
to 1, and two complex conjugate eigenvalues.
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The real eigenvalue corresponds to the eigenvector
e3, which is sent to itself by the rotation,
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hence the eigenvalue of 1.
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The fact that the other two eigenvalues are
complex means that no other vector is sent
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to a REAL scalar multiple of itself.
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This makes sense geometrically because NO
other vector points in the same direction
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it started in after being rotated. Now, how
do we describe any general rotation about
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an arbitrary axis?
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Well, a matrix is completely defined by how
it acts on basis vectors.
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Since a rotation doesn't change the lengths
of vectors or the angles BETWEEN two vectors,
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a rotated basis will also be a basis for R3!
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This tells us that any rotation matrix can
be described as an orthonormal matrix.
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The columns are the vectors each standard
basis vector is sent to.
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Is it true that ALL orthonormal matrices rotate
vectors? Pause the video. Nope, here's an
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orthonormal matrix that's not a rotation;
it's a reflection.
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The rule is that only an orthonormal matrix
whose determinant is positive 1 is a rotation.
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But let's get back to thinking about rigid
body KINEMATICS. Remember, we want to describe
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the motion of the disk. We've talked about
rotation matrices, but we've left out a very
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important component: time!
How will we describe time dependent rotation?
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That's right, time dependent matrices.
Let's start by modeling a simple motion: the
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rotation of a disk as it spins clockwise about
the positive z-axis.
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We know how to write a matrix that describes
rotation by an angle theta about the z-axis.
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How would you make this rotation time dependent?
Pause the video and discuss.
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The obvious choice here is to simply make
theta a function of time! But how does it
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depend on time?
To write an explicit function, we need to
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know the rate, omega, at which the disk is
rotating.
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Assume the disk spins with constant angular
velocity.
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We can easily calculate omega by counting
the revolutions per second.
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And there's our matrix for a spinning—but
not wobbling—disk.
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Now let's try a slightly more difficult example.
Let's describe the motion of this wobbly,
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spinning disk as it rotates on this stick.
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The disk is itself rotating clockwise about
its center of mass when viewed from the positive
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z-axis. As before, we can find the rotation
rate, omega-D, of a marked point by counting
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the revolutions per second. Assume omega_D
is constant. Now, observe the slight tilt
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of the disk off of horizontal.
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This tilt is created by a rotation about a
tilt axis.
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The tilt axis is the vector in the xy-plane
about which the disk is rotated by some small
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angle theta, creating the tilt. The wobble
is created because the tilt axis is rotating
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clockwise about the positive z-axis.
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We can visualize this by observing that the
normal vector to the disk rotates in a cone
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shape about the z-axis. By tracking the normal
vector's revolutions per second, we can find
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the rotation rate of the wobble, omega-W,
of the normal vector. This is also the rotation
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rate for tilt axis. We assume omega_W is constant.
Notice that omega-D and omega-W are different
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rotation rates.
For simplicity, we assume that the marked
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point begins along the x-axis; and the initial
tilt axis aligns with the x-axis, with the
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tilt angle theta.
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Let's start by creating a sequence of rotations
that rotates the marked point to the angle
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omega_D times t and the tilt axis to the angle
omega_W times t for any time t.
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To describe this motion, we are going to decompose
the behavior into a sequence of rotations
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about e1, e2, and e3, which have the benefit
of being easy to describe mathematically.
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I want to start by eliminating the tilt of
the disk, so we can imagine it spinning parallel
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to the ground.
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What is the matrix that undoes the tilt of
theta degrees about the x-axis? Pause and
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write down a matrix. We rotate by an angle
negative theta about the positive x-axis,
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which is represented by this matrix.
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Now, I rotate the marked point clockwise about
the z-axis by the angle (omega_D minus omega_W)
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times t.
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This matrix describes the angle difference
traveled by the marked point relative to the
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position of the tilt axis. Now, we need to
make sure that we tilt the disk again so that
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we can describe the wobble.
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Since we assume the tilt axis begins along
the x-axis, we rotate the disk back to the
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initial tilted position by theta degrees.
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This counterclockwise, time-independent rotation
about the x-axis is represented by this matrix.
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Finally we must describe the wobble, created
by the rotation of the tilt axis. The tilt
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axis is rotating clockwise about the positive
z-axis with rotation rate omega-W, so at time
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t, it has rotated by omega_W t degrees, which
is what this matrix does.
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How will we combine these matrices to describe
the motion of the marked point?
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Pause and discuss.
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We multiply the matrices together. The order
we apply each matrix matters. We must perform
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the rotations in the same order we decomposed
the motion, because matrices do not multiply
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commutatively.
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Let's understand geometrically why this worked.
The angle of the marker is changed in two
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steps of this process, first a rotation by
angle omega-D minus omega-W times t, and then
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by an angle omega_W times t.
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In the end, it ends up exactly where it should,
at omega-D times t. Only the final matrix
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affects the tilt axis, rotating it by the
angle omega-W times t.
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Because the disk is a rigid object, by describing
the position of the marked point and the tilt
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axis for all times with matrices, we've actually
described the position of every point on the
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disk.
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We can find the location of any point at time
t by applying this matrix operation to any
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vector on the initial disk. Now, let's go
back to the badly thrown disk.
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We can apply the rotational transformation
directly to our thrown disk.
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The only changes might be to the rotation
rates and the initial position. You can think
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about how we might change the formula. We'll
ignore that. So all that is left to consider
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is the translation.
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If you throw a disk and watch it from the
side, we can ignore the rotations and focus
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on the translation of the center point of
the disk. For the small time interval that
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we are interested in describing, the disk
moves in a straight, horizontal path. So this
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vector equation approximates the translation.
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Because the disk is a rigid object, we get
the full description of the motion by simply
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adding in the translation.
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To the rotation of the wobbly disk to obtain
the following equation of motion for any point
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on the disk.
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If you thought this problem was cool, you're
not the only one. Richard Feynman studied
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the kinematics AND the dynamics of the wobbly
disk. He was able to show that the rotation
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rate of the special marked point, omega-D,
was exactly twice the rotation rate, omega-W,
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of the tilt axis. This realization ultimately
led to insights into the behavior of electrons.
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In this video, we saw that rotation matrices
are orthogonal matrices with determinant equal
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to positive 1.
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The kinematics of rigid bodies involves breaking
a problem into translation and rotation.
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The rotations may be decomposed into several
time dependent rotation matrices that are
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multiplied together.
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The matrix product added to the translation
describes the location at any time of all
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points on the rigid body.
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I hope you'll try to describe the motions
of various rotating objects that you encounter.
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Have fun, and good luck!