Transcript Rotational Mechanics Notes

Bell Ringer
• In terms of
energy, what
happens to the
energy of an
object in free-fall?
Bell Ringer
• Identify one station from
yesterday’s lab.
–How did you change
the object’s center of
mass?
–What did this do to the
object?
Center of
Mass
Center of Gravity (CG)
• The point at the center of an
object’s weight distribution
– The force of gravity can be
considered to act on an object at
this one point
• Located in the geometric center
of a symmetrical object
• Located in the heavier end of
an asymmetrical object
• May be located where there is
no actual material
– i.e. A donut
Center of Gravity (CG)
• An object which
rotates/revolves about its
CG moves smoothly
• An object which
rotates/revolves about a
point other than its CG
tends to wobble
Center of Mass (CM)
• The average position of all of
the particles making up an
object
• Usually located very close to
the center of gravity
– Not the case with very tall
objects
– Gravity pulls harder on the
bottom of the object
making the bottom
“heavier”
Stability
• An object will topple if its CG
is tipped beyond its support
base
• An object is most stable
when its CG lie below their
support points
– Extra work is required to
lift the CG to the point of
toppling
Stability
Stability
• Three types of equilibrium:
– Stable: Object balanced
so that any motion will
raise its CG
Stability
• Three types of equilibrium:
– Unstable: Object balanced
so that any motion will
lower its CG
Stability
• Three types of equilibrium:
– Neutral: Object balanced
so that any motion will not
change location of CG
CG of People
• Men tend to have slightly
higher centers of gravity
than women
Bell Ringer
• This is a view of
the tennis ball
from above.
• Write down the
letter of the
correct path of
travel as the
string is cut.
• DO NOT
DISUSS WITH
THOSE
AROUND YOU
Bell Ringer
• Why, when
carrying a large
object, do you
tend to learn
back?
Rotational
Mechanics
Rotational Motion
• Axis: The straight line around
which circular motion takes
place
• Rotation: When an object
turns about an internal axis
– The Earth rotates on its axis
• Revolution: When an object
turns about an external axis
– The Earth revolves around
the Sun
Rotational Motion
• Period (T): The time it takes
for an object to complete one
full circle
– Units: seconds
Two Types of Velocity
– Tangential Velocity (v): The
speed of an object moving
along a circular path
• Units: m/s
• Direction of motion:
–Always changing
–Always tangent to the
circle
Two Types of Velocity
v = (2pr)/T
T = period of motion (s)
r = radius of circular path (m)
p = 3.14
Two Types of Velocity
v = (2pr)/T
T = period of motion (s)
r = radius of circular path (m)
p = 3.14
Two Types of Velocity
– Rotational Velocity (w): The
number of rotations or
revolutions per unit time
• Units:
– radians per second
– revolutions per minute (rpm)
• i.e.) All parts of a turntable
have the same rotational
velocity
Relating Tangential and
Rotational Velocity
v = rw
• Therefore, for any
rigidly rotating system:
– All parts have the same
rotational speed
– Tangential speed
depends on rotational
speed, w, and radius, r
Bell Ringer
• If a car does doughnuts
with a radius of 6 m and
completes one full circle
every 3s, what is the
car’s tangential
velocity?
• What is the car’s
rotational velocity?
Which is the right path?
• This is a view of
the tennis ball
from above.
• Write down the
letter of the
correct path of
travel as the
string is cut.
Why does a
CD case slide
across the
dashboard on
a turn?
What really happens…
What is the right answer?
What is a piece of equipment
that use centripetal force for a
mechanical advantage?
Centripetal Force
• The force that causes an
object to follow a curved
path
– “Center-seeking” force
– Always directed at a
right angle to the
direction of motion
Fc =
2
(mv )/r
Centripetal Force
If the centripetal force stops
acting, the object will fly off in
a straight line path, tangent to
the circle
Centripetal Acceleration
• The change in velocity of
an object in rotational
motion, caused by the
centripetal force
ac = v2/r
– Occurs even if tangential
speed remains constant
– Object is still changing
direction to maintain its
circular path
CentriFUGal Force
• “Fictional” center-fleeing
force
• Only felt by an object within
a rotating reference frame
– Simply a reaction to the
centripetal force
ACTION
REACTION
CentriFUGal Force
CentriFUGal Force
Action: Centripetal force pushes object
into the circle
Reaction: Object exerts centrifugal force
back on the surface away from the circle
Simulated Gravity
• Comes from the centrifugal
force acting on rotating object
• Simulated gravity will feel
stronger when:
– The capsule spins faster
– The object sits farther from
the axis of rotation of the
capsule
• If sitting on the axis of rotation,
the object will feel no “gravity”
Bell Ringer
A friend pushes you
out of his car as
you go around a
turn.
In what direction will
you travel?
Bell Ringer
• What provides the
centripetal force for a
tetherball?
• What is the only way
to measure the
centrifugal force?
Bell Ringer
• In order for there to be a
“simulated gravity” effect, what
must happen?
• (in other words what must
something equal to)
• Examining G Forces
Torque
• Recall: Forces tend to
make objects accelerate
• Torque makes an object
rotate
• Torques occur when a
force is applied with
leverage
– Note: A perpendicular push
or pull gives more rotation
with less effort
Torque
• When the applied
force is perpendicular:
–Force is represented
by F
–Lever arm (l): the
distance from the axis
to the point of contact
Torque
• Therefore, torque (t) is:
t = Fl
– Units: Newton meters (N.m)
• The same torque can be
produced with:
– Large force & small lever
arm
– Small force & large lever
arm
Torque
• If a force is applied
directly to the CG (l = 0),
no rotation will occur
–Kicking a football directly
in its CG (no rotation) vs.
off its CG (rotation)
Balanced Torques
• In order for an object to
remain balanced the
torques on either end must
balance each other
• Therefore:
tccw = tcw
– ccw = counter-clockwise
– cw = clockwise
Balanced Torques
Bell Ringer
• What are three
things that you
can do to increase
the amount of
torque on a
stubborn rusted
bolt?
Rotational Inertia
• Recall: Newton’s Law of
Inertia
• There is a similar Law of
Rotational Inertia:
An object rotating about an
axis tends to keep rotating
about that axis unless acted
upon by a net torque.
Rotational Inertia
• Rotational inertia (I)
depends on the
distribution of mass of an
object
– Objects with mass far from
their CGs will have more
rotational inertia
Rotational Inertia
Rotational Inertia
• Which has less rotational
inertia:
– A short pendulum or a
long pendulum of the
same mass?
• Short pendulum
– A hoop or a solid disk of
the same mass?
• Solid disk
Rotational Inertia
Bell Ringer
• What are three
things that you
can do to increase
the amount of
torque on a
stubborn rusted
bolt?
Rotational Inertia
Angular Momentum
• Recall: Any moving object
has momentum
• Similarly, any rotating object
has angular momentum
Angular Momentum = Iw
• The more angular
momentum an object has,
the more torque required to
change it
– i.e. moving bicycle vs.
stationary bicycle
Angular Momentum
Angular Momentum
Conservation of Angular
Momentum
• Law of Conservation of
Angular Momentum:
Without a balanced, external
torque, the angular
momentum of a system will
remain constant
• In an isolated system:
– If I increases, w will decrease
– If I decreases, w will increase
Conservation of Angular
Momentum
Conservation of Angular
Momentum