Transcript Rotational Motion and Equilibrium

Rolling
Motion
of a Rigid
Object
AP Physics C
Mrs. Coyle
For pure rolling motion there is
“rolling without slipping”, so at
point P vp =0.
• All points
instantaneously
rotate about the
point of contact
between the
object and the
surface (P).
vp’ = 2 vcm
Rolling Motion
vCM 
a CM 
ds
 R
d
dt
dt
dv C M
d
dt
 R
dt
 R
 R
Speed and Acceleration of the CM
of a Rolling Object
vcm = ωR
acm = α R
Red Line: Path of a particle on a
rolling object (cycloid)
Green line: Path of the center of mass
of the rolling object
http://cnx.org/content/m14374/latest/
Rolling Motion: a combination of
pure translation and pure rotation.
The Total Kinetic Energy of a
Rolling Object is the sum of the
rotational and the translational
kinetic energy.
K = ½ ICM
2
ω
+ ½ MvCM
2
Note
• Rolling is possible when there is
friction between the surface and the
rolling object.
• The frictional force provides the
torque to rotate the object.
Ex: Accelerated Rolling Motion
Ki + Ui = Kf + U f
Mgh = ½ ICM ω2 + ½ MvCM2
vcm = ωR
There is no frictional
work. Why not?
Does friction cause a
displacement at its
point of action?
Ex: #52
A bowling ball (on a horizontal
surface) has a mass M, radius R, and a
moment of inertia of (2/5)MR2 . If it
starts from rest, how much work must
be done on it to set it rolling without
slipping at a linear speed v? Express
the work in terms of M and v.
Ans: (7/10)Mv2
Ex: #54
• A uniform solid disk and a uniform hoop are
placed side by side at the top of an incline of
height h. If they are released from rest and roll
without slipping, which object reaches the
bottom first? Verify your answer by calculating
their speeds when they reach the bottom in
terms of h.
• Ans: The disk, vdisk =(4gh/3)1/2 , vring =(gh)1/2