Transcript rotational inertia

8 Rotational Motion
• describing rotation & “2nd Law” for rotation
• conserved rotations
• center of mass & stability
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Homework:
RQ: 1,2, 6, 7, 13, 14, 16, 18, 20, 23, 32.
Ex: 22, 40, 47.
Problem: 3.
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Describing Rotation
• speed on rotating object ~ distance from
center (v ~ r)
• however, all points on object have same
#rotations/second (v/r is same)
• rotational speed = v/r
• “direction”: clockwise or counter-clockwise
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torque
• torque = force x lever-arm
• lever-arm determined by force-line as
shown below
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torque examples
• large torque:
• lever-arms = r
• zero torque:
• lever-arms = 0
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rotational inertia
• translational inertia depends only on the
mass of the object
• rotational inertia depends on mass,
shape, and size.
• bigger objects tend to have higher
rotational inertias
• e.g. tightwalker uses long pole for
balance.
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high rotational inertia
• improves stability of rotating objects
• Examples: turntable platter, bicycle wheel,
frisbee
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Example Torque
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Meter stick balanced from 50cm
Hang 200grams from 30cm
Lever-arm = 20cm
?Torque?
Force x lever-arm = weight of 200 grams x
20cm ~ 200grams x 20cm = 4000gram-cm
• Where can we put 100grams so that it
balances the meter stick?
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Pract. Phys. p.42 #2.
• 2a) twice the lever-arm, W = 250N
CCW-torque = 250Nx4m = 1000mN
CW-torque = 500Nx2m = 1000mN
• Rotational Equilibrium is state when CCW
and CW torques are equal.
• We also say that the “sum of the torques”
is zero for Rot. Eq.
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Cont.
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2b) 4/3 weight ratio, what is lever-arm?
CCW-torque = 300Nx4m=1200Nm
CW-torque = 400Nx??m = 1200Nm
=> 3m balances the see-saw.
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Cont.
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2c) weight of board balances child.
CCW-torque = 600Nx1m = 600Nm
CW-torque = ??Nx3m = 600Nm
??N = 600Nm/3m = 200N
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angular momentum
• linear momentum:
(linear inertia) x (linear velocity) = mv
• angular momentum:
(rotational inertia) x (rotational velocity)
• conserved when net external torque is zero
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isolated rotation
• (rotat. inertia) x (rotat. velocity) = constant
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rotational velocity ~
rotational inertia
• skater, diver, gymnast “tuck” to spin faster,
“open” to spin slower
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center of mass
• average location of object’s mass
• at center of symmetrical objects (hoop,
ball, rod)
• affects stability (rollover risk ~ height of
center of mass)
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center of mass
• closer to more massive object
• equal to “center of gravity” (for small
system)
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stability
• tendency to maintain position
• center of mass must be above base
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wine rack
• where is the center of mass?
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equilibrium
• condition of Fnet = 0 & net torque = 0
• stable objects are in equilibrium
• not all equilibriums are stable
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centripetal force
• force required to maintain curved motion
• points toward center of curvature
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centrifugal force (p44 #1, #3)
• not a force on object
• opposite of centripetal
• apparent force due
to inertia
• e.g. ball (or water) in a
spinning bucket “feels”
a force holding it
to the bottom.
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Practicing Physics
• P46 circular motion
• P48 rotational dynamics
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Summary
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angular velocity = v/r
torque = force x lever-arm
rotational inertia = ‘rotational mass’
angular momentum of isolated systems is
conserved
• definitions: center of mass, stability,
equilibrium
• centripetal & centrifugal forces
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