Download Tangential velocity Angular velocity

Concept Summary
Belton High School Physics
Circular Motion Terms
• The point or line that is the
center of the circle is the axis
of rotation.
• If the axis of rotation is inside
the object, the object is
rotating (spinning).
• If the axis of rotation is outside
the object, the object is
revolving.
Uniform Circular Motion
• The motion of an object in a circle
traveling with uniform or constant
speed.
• Velocity is constantly changing.
In circular motion this
velocity becomes a little
more complex. We now
have 2 types of velocity…
•Tangential velocity
•Angular velocity
Linear/Tangential Velocity
v
• Objects moving in a circle still have a
linear velocity = distance/time.
• This is often called tangential velocity,
since the direction of the linear velocity
is tangent to the circle.
v
2 pod racers complete a turn and
remain neck and neck. Which had the
greater Linear Velocity?
A. The one closer to the point of rotation.
B. The one farther from the point of rotation.
C. Both were equal.
Angular Velocity
w
• rotational or angular
velocity, which is the
rate angular position
changes.
• Rotational velocity is
measured in
degrees/second,
rotations/minute
(rpm), etc.
• Common symbol, w
(Greek letter omega)
Rotational/Angular Velocity
• Rotational velocity =
Notice that we don’t
have to use angles!
We could just as
easily use Radians!
Change in angle
time
How would you describe the angular
velocity of the 2 pod racers in the
previous question?
A. Greater for the one closer to the point
of rotation.
B. Greater for the one farther from the
point of rotation.
C. Both the same.
Rotational Vs. Tangential
Velocity
• If an object is rotating:
– All points on the object have the same
rotational (angular) velocity.
– All points on the object do not have the
same linear (tangential) velocity
Rotational & Tangential Velocity
• We now see that….
– Tangential (linear) velocity of a point
depends on:
• The rotational velocity of the point.
– More rotational velocity can mean more linear
velocity.
– The distance from the axis of
rotation.
• More distance from the axis means more
linear velocity
Analogies Between Linear and
Rotational Motion
Relationship Between Angular and
Linear Quantities
• Displacements
d  r
• Speeds
vt  w r
• Accelerations
at   r
• Every point on the
rotating object has
the same angular
motion
• Every point on the
rotating object does
not have the same
linear motion
Acceleration
• As an object moves around a circle, its
direction of motion is constantly
changing.
• Therefore its velocity is changing.
• Therefore an object moving in a circle
is constantly accelerating.
Centripetal Acceleration
• The acceleration of an object moving in
a circle points toward the center of
the circle.
• This is called a centripetal (center
pointing) acceleration.
a
Centripetal Acceleration
• The centripetal acceleration depends
on:
– The speed of the object
– The radius of the circle
acent =
2
v
r
Centripetal Acceleration and
Angular Velocity
• The angular velocity and the linear
velocity are related (v = ωr)
• The centripetal acceleration can also be
related to the angular velocity
v
r ω
2
aC  
 rω
r
r
2
2
2
Centripetal Force
• Newton’s Second Law says that if an object
is accelerating, there must be a net force on
it.
• For an object moving in a circle, this is called
the centripetal force.
• Newton’s Second Law also states that this
net force must point in the same direction
as the acceleration.
• The centripetal force points toward the
center of the circle.
Centripetal Force
• In order to make an object revolve
about an axis, the net force on the
object must pull it toward the center
of the circle.
• This force is called a centripetal (center
seeking) force.
Fnet
Centripetal Force
• Centripetal force on an object depends
on:
– The object’s mass - more mass means
more force.
– The object’s speed - more speed means
more force
– And…
Centripetal Force
• The centripetal force on an object also
depends on:
– The object’s distance from the axis
(radius)
• If linear velocity is held constant, increasing
the radius requires decreases this force.
• If rotational velocity is held constant,
increasing the radius increases the force.
Centripetal Force
In symbols: F= ma so……
So….
• Since ac= v2/r
Fc = mv2/r
• And v= ωr
2r
2
F
=
mω
• Then ac = ω r
c
“Centrifugal Force”
• “centrifugal force” is a fictitious force it is not an interaction between 2
objects, and therefore not a real force.
• Nothing pulls an object away from
the center of the circle.
“Centrifugal Force”
• What is erroneously attributed to
“centrifugal force” is actually the action
of the object’s inertia - whatever
velocity it has (speed + direction) it
wants to keep.
A note on the Vector Nature
of Angular Quantities
• Angular displacement,
velocity and acceleration
are all vector quantities
• Direction can be more
completely defined by
using the right hand rule
– Grasp the axis of rotation
with your right hand
– Wrap your fingers in the
direction of rotation
– Your thumb points in the
direction of ω
In example (a) the direction
of angular velocity ω points
into the page.
A. True
B. False
If the rotational speed of a space
station increases then the apparent
weight of its inhabitants will
increase.
A. True
B. False
An astronaut decides to go jogging in his
relatively small rotating space station. If
wants to burn the most calories he should
A. Jog in the opposite direction
that the station is rotating.
B. Jog in the same direction that
the station is rotating.
C. Run in place at the very
center of the station.
Circular motion Problem Solving
Strategy
• Draw a free body diagram, showing and
labeling all the forces acting on the
object(s)
• Find the net force toward the center of
the circular path (this is the force that
causes the centripetal acceleration, FC)
• Apply Newton’s 2nd.
• Solve for unknowns.
Common situations involving
Centripetal Acceleration
• Many specific situations will use forces
that cause centripetal acceleration
– Level curves
– Banked curves
– Horizontal circles
– Vertical circles
• Note that Fc, v or ac may not be constant
Level Curves
• Friction is the force
that produces the
centripetal
acceleration
• Can find the
frictional force, µ, or
v
v  rg
Banked Curves
• A component of the
normal force adds
to the frictional
force to allow
higher speeds
v2
tan  
rg
or ac  g tan 
Vertical Circle
• Look at the forces at
the top of the circle
• The minimum speed
at the top of the circle
can be found
v top  gR
The End