Transcript Angular Mechanics - seniorphysicscranson

Angular Mechanics
Chapter 8/9
Similarities
Linear
Angular
Mass
Moment of Inertia
Force
Torque
Momentum
Angular Momentum
Center of Mass
• The center of mass of an object is the
average position of mass.
• Objects tend to rotate about their center of
mass.
• Examples:
• Meter stick
• Rotating Hammer
• Rolling Double-Cone
Stability
• For stability center of gravity must be over area
of support.
• Examples:
• Tower of Pisa
• Touching toes with back to wall
• Meter stick over the edge
Otherwise we will get a rotation!
I = Rotational Inertia
• An object rotating about an axis tends to
remain rotating unless interfered with by
some external influence.
• This influence is called torque.
• Rotation adds stability to linear motion.
– Examples:
• spinning football
• bicycle tires
• Frisbee
Moment of Inertia
• Defined as
resistance to
rotation
– depends on mass
– depends on
distance from axis
of rotation
I = mr2
Moments For Various Objects
Object
Location of
axis
Thin Hoop
Center
mr 2
Solid Cylinder
Center
Uniform
Sphere
Center
1 2
mr
2
2 2
mr
5
Uniform Rod
Length L
Uniform Rod
Length L
Center
1
ml 2
12
Through End
1 2
ml
3
Thin Plate
Length L Width
W
Center
Diagram
Moment of
Inertia
1
m(l 2  w 2 )
12
• The greater the distance between the
bulk of an object's mass and its axis of
rotation, the greater the rotational
inertia.
• Examples:
– Tightrope walker
– Metronome
Ways to Measure Rotation
• Degrees: 1/360th of a
revolution
1
• Radians:  of a
2
revolution
1 revolution = 2  radians
Angular Displacement
• Found by change in θ.
• Distance around a pivot is
found by
• d=rθ
– Where the angle is measured in
radians and r is the radius of the
arc.
– Measured in meters
Angular Velocity
• The rate of revolution around an axis.
– Measured in rads/sec


t
• Velocity around an axis is found by
v = rω
Where r is the radius and ω is angular velocity and
is measured in m/s.
How Fast Does the Earth Spin?
• 1rev/24 hrs
• 2π radians/revolution
• Radius or earth = 6.38x106m
1rev
2 rad
5 rad
e 

 7.27x10
24hrs 86400s
sec
rad
m
v  r  6.38x10 m(7.27x10
)  464
s
s
6
5
Angular Acceleration
• The change in angular velocity per unit
of time.
– Measured in rads/sec2


t
Acceleration of an object is found by
a = rα
And is measured in m/s2.
Linear and Angular Measures
Quantity
Linear
Angular
Relationship
Displacement
d (m)
θ (rad)
d=rθ
Velocity
v (m/s)
ω (rad/s)
v=rω
Acceleration
a (m/s2)
α (rad/s2)
a=rα
Centripetal or Centrifugal?
Direction of
Motion
Centripetal
Force
Centrifugal
Force
No Matter What Faith Hill Says,
IT’S NOT CENTRIFICAL MOTION!
Centripetal Force
• …is applied by some object.
• Centripetal means "center seeking".
Centrifugal Force
• …results from a natural tendency.
• Centrifugal means "center fleeing".
• This is a fictitious force for us. Why?
Centripetal motion
2
v
ac 
r
mv
Fc 
r
Practice Wall and Wall
2
Pg 234, 6-6,
Pg 243, 6-12
Examples
Centripetal Centrifugal
Force
Force
• water in bucket
• Bucket
• Nature
• moon’s orbit
• Earth’s gravity
• Nature
• car on circular path
• Road Friction
• Nature
• coin on a hanger
• Hanger
• Nature
• jogging in a space
station
• Space Station
Floor
• Nature
Conservation of Angular
Momentum
• angular momentum = rotational inertia ´ rotational
velocity
•
L=I
• Newton's first law for rotating systems:
– “A body will maintain its state of angular momentum
unless acted upon by an unbalanced external torque.”
• Examples:
–1. ice skater spin
–2. cat dropped on back
–3. Diving
–4. Collapsing Stars (neutron
stars)
Torque
• Force directed on an object that has a
fixed point is found by
  Frsin
– Where τ is torque, F is force in N, r is
distance from the axis in m, and θ is
measured IN DEGREES.
– (use sin θ only if force is not || to motion)
Levers
Lever arm
(r sin θ)
• The lever arm is the
distance from the
axis along a θ to the
direction of applied
force.
• Torque here is force
times the lever arm.
  Frsin
r
θ
A Balancing Act
• Static equilibrium occurs when the sum
of the torques add to equal zero.
NOW BUILD YOUR OWN!
• Must involve at least 3 different axes of
rotation.
• Must hang at least 8 objects.
• No 2 objects can have the same mass.
• No two hangers can have the same length.
• No fulcrum can be in the middle of a hanger.