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Physics 211. 10: Angular Momentum and Torque. Rolling Motion of a Rigid Body Vector Product and Torque Angular Momentum Rotation of a Rigid Body about a Fixed Axis Conservation of Angular Momentum. Rolling Motion of a Rigid Body. Rotation + Translation. If v. =. r. w. cm. ß. - PowerPoint PPT Presentation
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Physics 211
•Rolling Motion of a Rigid Body•Vector Product and Torque•Angular Momentum•Rotation of a Rigid Body about a Fixed Axis•Conservation of Angular Momentum
10: Angular Momentum and Torque
Rolling Motion of a Rigid Body
Rotation + Translation
If v
cm r
Pure Rolling Motion
As then one can see that the center of mass moves at v cm
by noting that 2f angular frequency=angular speed
where f frequency of rotation, which has unitsrps=revolutions per second
or cps =cycles per second
Which shows that the distance traveled by the center ofmass in pure rolling motion in one second is
2r f2r2
r
vcm
vtop = vcm+ v
vbottom = vcm- v
Pure rolling motion = no skidding
v = linear speed of rim of wheel just due to the rotational motion; vtop & vbottom are the total linear speeds at the top and bottom
v = vcm thus for pure rolling motionvtop = vcm+ vcm= 2vcm
vbottom = vcm- vcm= 0!!!!!!
Skidding = no rotational motion about of center of mass
Kinetic Energy of moving wheelKtot Krot Ktrans
Ktot 12I cm
2 12Mvcm
2
N
W
Fs
No work done by non conservative forces in pure rolling motion
tot.mech=
Ktot= - Utot
h
Utot = -mghKtot = Ktot,final - Ktot,initial = Ktot,final (as initially at rest)Ktot = Krot+ Ktrans
As there is no slipping (skidding) the rolling object only experiences STATIC friction with the surface
Static friction can NEVER do any work
At each instant of time the static friction stops translational motion and causes rotation
[The static friction does not have to be the maximum possible static friction (i.e. it can be ]
The static friction produces an instantaneous torque on the portion of the object in contact with the surface
This torque does NO work as it is only in contact with the SAME portion of the object for an infinitesimal time.
)s N
Vector Producta X b aa b b
a b a b sin
is the angle between a and b
X
X
axb
b
a
b
a
axb
Right Hand Rule
Vector Product
a b i j kax ay azbx by bz
b a
a ybz azby i a xbz azbx j axby aybx k
i j i j k1 0 00 1 0
k
j k i j k0 1 00 0 1
i
i k i j k1 0 00 0 1
j
j
k
i
right handed axis system
Using the vector product Torque can be written as rF
r = position vector from an origin to the point of contact of the force F
unit vector in direction of rotational axis is
ˆ r ˆ F =ˆ
F
rˆ r ˆ F ˆ
Choose an origin then draw the position vector
If F is the total force then is the total
torque
Properties of Vector Producta X b absin
aa X b & ba ba a 0
a b b addt
a b dadt
b a dbdt
X
X
X
X
X X X
As 0
ddt
ddt
ddt
pτ r F r
r p
r p v p
Angular Momentum
L rpL depends on the choice of origin
L rp rpt
L rptrmvtrmr mr 2 I L I ̂
rpt
ddt
ddt
τ r p
Lτ r F
.Tot
Tot cm Tot cm Tot External
Tot kk
Tot kk
ddt
Lτ r F r F
τ τ
L L
Forces acting on many particles that are rigidlyfixed with respect to each other or an
extended rigid body
Thus if only internal forces act
tot 0 int
In general for any system made upof many objects that are fixed with respect to each other or an extended rigid object
totint extext
The vector product and the choice of origin
defines the axis of rotation of the rigid body
.
is the angular acceleration of the center of mass
about the axis determined by thechoice of origin
TotTot cm Tot cm Tot External
cm
ddt
I
Lτ r F r F
αα
Conservation of Total Angular Momentum
If a system is isolated
tot
0 dL
tot
dtConservation of Total Angular Momentum
L tot,initial L tot,final
e.g.
I1i 1i ˆ I2i 2 i ˆ I1 f 1 f ˆ I2 f 2 f ˆ