Professor Dave here, let's discuss angular motion.
We talked about uniform circular motion, but we
need to make an important distinction
between spin and orbital motion.
An object can spin around an internal axis
that goes through its center of mass, and
it can also orbit around some external
axis. This means that the earth will spin
on its axis but it orbits around the Sun.
We have discussed spinning objects and
the way that tangential speed will vary
according to distance from the axis of
rotation, but this can be a tricky way to
view the rotation of something like a
ferris wheel, because there is a certain
context in which we would like to say
that every part of the wheel is spinning
at the same speed, since it is one solid
object. For this reason when we are
looking at a rotating rigid object,
rather than looking at translational
motion, we will want to discuss angular
motion or rotational motion. To do this
we can begin to discuss certain angular
quantities. Angular displacement
represented by theta, will be the angle
swept out by any line passing through a
rotating body that intersects the axis
of rotation. This value will be positive
if the motion is counterclockwise and
negative if clockwise. If you have a hard
time remembering this just remember that
in going counterclockwise around the XY
plane we will cover quadrants one, two,
three, and then four, so positive angular
displacement correlates with a positive
increase in the number quadrant that is
being traversed over time. The SI unit
for angular displacement will not be
degrees but rather radians. We will
discuss the derivation and importance of
the radian in the upcoming mathematics
course, but for now we simply need to
know that one full revolution is equal
to 2 pi radians, so if the rotation
completed by the object is a
counterclockwise quarter turn, the
angular displacement is
one half pi radians, or pi over two.
Angular velocity, as one might expect, is
equal to the angular displacement over
some time period, just the same way that
linear velocity is equal to linear
displacement over time. Average angular
velocity, represented by the Greek letter
omega, is therefore given as delta theta
over delta t. The SI unit for this value
will be radians per second, although we
will frequently encounter other units
like revolutions per minute, or rpm. Again,
counterclockwise rotation involves
positive angular velocity and clockwise
will be negative. Lastly, angular
acceleration, represented by the Greek
letter alpha, is equal to the change in
angular velocity over some time period.
This will be equal to delta omega over
delta t, with units of radians per second
squared. Now that we have defined these
terms, we can begin to understand that
rotational kinematics will utilize
equations that are direct analogs of
those involved with linear kinematics.
We just swap out linear components for
angular ones. Looking at these familiar
equations, if we just exchange
displacement for angular displacement,
velocity for angular velocity, and
acceleration for angular acceleration, we
arrive at a new set of equations that
allow us to calculate rotational motion,
and they will apply to any system with
rotational motion under constant angular
acceleration. Depending on the scenario
it may or may not be easier to model
rotational motion with these equations
rather than with the linear ones where
we have to use tangential velocities
that vary according to the radius.
How can we set a system into rotational
motion? We can do this by generating
something called a torque. Torque is
defined as the ability of a force to
rotate an object around some axis,
whether we are looking at a wheel, a
seesaw, or anything else that exhibits
this kind of motion, and for now we can
just look at a top-down view of a panel
attached to a rod allowing for free rotation.
Torque, represented by the Greek letter
tau, will be equal to the applied force
times the distance over which the force
is applied times sine theta, where theta
is the angle between the force vector
and the plane of the rotating object.
Torque will therefore have units of
Newton meters, and torque will be at a
maximum when the applied force is
perpendicular to the plane containing
the object, because the sine of 90
degrees is one. Any angle less than 90 in
either direction will give a sine value
that is less than 1, and will therefore
diminish the torque, while still
producing some motion, until the angle is
zero, at which point the whole expression
will be equal to zero, and no motion can
result. By convention, we will say that
torque is positive if it results in
counterclockwise rotation and negative
if it results in clockwise rotation, as
we are by now familiar. This must be the
case, because if two different forces act
upon this object to produce opposite
torques that are equal in magnitude, the
object will not move, and the sum of the
individual torques must be zero to
reflect this fact. We must also note that
the magnitude of torque will be equal to
the magnitude of the applied force times
the length of the lever arm expressed in
Newton meters. The longer the lever, the
easier it is to produce torque, which is
why it is difficult to push a door open
if you press near the hinge, but easier
and easier as you move outwards, thus
extending the lever arm, or the distance
from the axis of rotation to the point
where the force is applied. This sheds
light on the quote by Archimedes, which
is generally reported as some variation
of the following: give me a lever long
enough and a fulcrum to place it on, and
I will move the earth. This implies that
with a long enough lever arm, one could
generate such a massive torque so as to
move the world. Wise words from a wise
man ,so let's learn about some other
things Archimedes said next, but first
let's check comprehension.
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