Rotational Motion and Equilibrium

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-Angular and Linear Quantities
-Rotational Kinetic Energy
-Moment of Inertia
AP Physics C
Mrs. Coyle
Tangential (Linear) Speed and
Angular Speed
s=qr
v 
ds
dt
 r
dq
 r
dt
v  r
Tangential Acceleration and
Angular Acceleration
at 
dv
dt
r
d
 r
dt
a  r
Angular and Linear Quantities
• Displacements
• Speeds
• Accelerations
s qr
v  r
a  r
• Every point on the rotating object has the
same angular motion but not the same linear
motion. The a and v are functions of r.
Remember:
Centripetal Acceleration
aC 
v
2
 r
r
v is the tangential speed
2
Resultant Linear Acceleration
• The net acceleration is
the sum of the
tangential and
centripetal
accelerations.
a
a a
2
r
2
t
Rotational Kinetic Energy
• A particle in a rotating object has rotational
kinetic energy:
Ki = ½ mivi2 , vi = i r (tangential velocity)
For the
Object:
KR 
K
i
KR
i

1
 2m r
i i
2

2
i
1
1
2 
2
2
   m i ri   I 
2 i
2

Rotational Kinetic Energy and
Moment of Inertia
• The total rotational kinetic energy of the rigid
object is the sum of the energies of all its
particles
1
2
2
K R   K i   m i ri 
i
i 2
KR
1
1
2 
2
2
   m i ri   I 
2 i
2

• I is called the moment of inertia
Moment of Inertia, I
• Moment of Inertia, I, is a measure of
the resistance of an object to
changes in its rotational motion.
• Moment of Inertia is analogous to
mass in translational motion.
Example #20
Rigid rods of negligible mass lying along the y
axis connect three particles. If the system
rotates about the x-axis with an angular speed
of 2.00rad/s find a) the moment of inertia about
the x-axis and the total rotational kinetic energy
evaluated from ½ I ω2 and b) the tangential
speed of each particle and the total kinetic
energy evaluated from ½ mi vi 2
Ans: a)92kg m2 , 184J , b) 6m/s,4m/s,8m/s,184J
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