Angular momentum is one of the more intriguing concepts
in physics and in many important situations it is a conserved quantity.
In particular, by analogy with CONP, the angular momentum of a system
is conserved whenever the net external torque on a system is zero. Also
by analogy with linear momentum (= mass (inertia) times velocity),
angular momentum equals moment of inertia (or rotational inertia) times
angular velocity. Yes folks, the analogy is complete. For most all of the
linear quantities that we have studied thus far, there exists a rotational
analog. We move from one place to another in a linear way all the time.
We're used to it since it's part of our everyday experience. However,
we rarely rotate - that's one of the reasons why amusement parks are amusing.
Today's hands-on activities are intended to give you some experience
with rotational quantities that are important to understanding angular
momentum. 1. GYROSCOPES AND THE NORDON BOMBSIGHTYou've probably played with toy gyroscopes before.
However, their unique properties make them useful for far more serious
applications than toys. Carefully observe the motions of these gyroscopes.
They "want" to maintain their orientations in space. Why they "want" to
do so is our main interest. Applications? - there are many! The Nordon
bombsight (actually an "autopilot" device) brings home the idea of the
cross product and the vector nature of quantities like torque (=r
X F) and angular mometum (=r X p) better than most
demonstration tools. Ever push on something and have it move at right
angles to the direction that you push it?2. AIR GYRO AND THE HANGING BICYCLE WHEEL - PRECESSIONThough they may "want" to maintain their orientation
in space, not all gyroscopes are able to do so. Why not? The ablility to
recognize when angular momentum is conserved and why it is conserved is
important. Why angular momentum is not conserved in some situations and
the effects of it not being conserved are also important. Precession is
one such effect. Carefully observe how the rate of precession (quantified
by precession angular velocity) of these gyros varies as functions
of the external torques on, and the angular momenta of, the gyros.
Try to relate the behavior of the hanging bicycle wheel to your experience
of learning how to ride a bike. Remember the advice that "it's easier
the faster you go." Recall when you learned how to "ride with no
hands." You probably thought "hey, I'm cool - look what a good sense
of balance I have." Sorry, the wheel was simply "reluctant" to fall
over.3. CONSERVATION OF ANGULAR MOMENTUM I: MAGNITUDEThe ice skater trick. This one shows the magnitude
of angular momentum being conserved. Not recommended after a big meal!4. CONSERVATION OF ANGULAR MOMENTUM II: DIRECTIONAngular momentum is a vector quantity. If
angular momentum is conserved, both the magnitude and the direction of
the angular momentum must remain constant. The system of the spinning bicycle
wheel, the student and the rotating platform the student stands upon has
its angular momentum conserved. Why? Hang on to your hat (and breakfast
too!).5. THE GREAT RACE: ROTATIONAL INERTIAOK, this one is a repeat. The BIG ANALOGY between
linear and rotational kinematics and dynamics is very useful. However,
analogies are rarely perfect. Mass (inertia - a linear dynamics concept)
and rotational inertia are certainly analogous. However, inertia
is inertia is inertia. Rotational inertia on the other hand depends
sensitively on the axis of rotation and the mass distribution of an object
about that axis of rotation. This is why your instructor has,
until now, insisted on talking about objects sliding down inclines
rather than rolling down inclines? This demo will show you why.
If the object also rotates, the effects of that rotation must be accounted
for. How does rotational inertia come into play in the "race?" See
how many concepts of linear dynamics can be extended by analogy
to the understanding of why the ring always loses the race. Try the
work-energy principle (WEP) and the principle of conservation of mechanical
energy (CONE). Hey, even try to puzzle out how Newton's Laws speak
to the reason why the disk wins the race down the incline.
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