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PHY 113 A Fall 2012 -- Lecture 17 110/10/2012
PHY 113 A General Physics I9-9:50 AM MWF Olin 101
Plan for Lecture 17:
Chapter 10 – rotational motion
1. Angular variables
2. Rotational energy
3. Moment of inertia
PHY 113 A Fall 2012 -- Lecture 17 210/10/2012
PHY 113 A Fall 2012 -- Lecture 17 310/10/2012
Angular motion
angular “displacement” q(t)
angular “velocity”
angular “acceleration” dtd
(t)ω
α dtd
(t)θ
ω
“natural” unit == 1 radian
Relation to linear variables: sq = r (qf-qi)
vq = r w
aq = r a
s
PHY 113 A Fall 2012 -- Lecture 17 410/10/2012
Special case of constant angular acceleration: a = a0:
(w t) = wi + a0 t
(q t) = qi + wi t + ½ a0 t2
( (w t))2 = wi2 + 2 a0 ( (q t) - qi )
w
r1v1=r1w
v2=r2w
r2
PHY 113 A Fall 2012 -- Lecture 17 510/10/2012
A wheel is initially rotating at a rate of f=30 rev/sec.
smRv
R
πfω
/247.94)0.5m)(rad/s 495.188(
0.5m? radius aat wheel theof
rim on thedot a of speed theisWhat
rad/s 188.495
rad/s )30(22
elocity?angular v theisWhat
R
PHY 113 A Fall 2012 -- Lecture 17 610/10/2012
A wheel is initially rotating at a rate of f=30 rev/sec. Because of a constant angular deceleration, the wheel comes to rest in 3 seconds.
22
2
/42.31)0.5m)(rad/s 83.62(
0.5m? radius aat wheel theof
rim on thedot a ofon decelerati theisWhat
rad/s 83.26
3
rad/s )30(2
3
20
on?deceleratiangular theisWhat
smRa
R
ss
πf
R
PHY 113 A Fall 2012 -- Lecture 17 710/10/2012
Example: Compact disc motion
In a compact disk, each spot on the disk passes the laser-lens system at a constant linear speed of vq = 1.3 m/s.
w1=vq/r1=56.5 rad/s
w2=vq/r2=22.4 rad/s
What is the average angular acceleration of the CD over the time interval Dt=4473 s as the spot moves from the inner to outer radii?
a = (w2-w1)/Dt =-0.0076 rad/s2
w1
w2
PHY 113 A Fall 2012 -- Lecture 17 810/10/2012
Object rotating with constant angular velocity (a = 0)
w
v=0
v=Rw
Kinetic energy associated with rotation:
“moment of inertia”
iii
iii
iii
rmI
IrmvmK
2
22
1222
122
1
:where
;ωω
R
PHY 113 A Fall 2012 -- Lecture 17 910/10/2012
Moment of inertia: i
iirmI 2
iclicker exercise: Which case has the larger I? A. a B. b
PHY 113 A Fall 2012 -- Lecture 17 1010/10/2012
Moment of inertia: i
iirmI 2
22MaI 22 22 mbMaI
PHY 113 A Fall 2012 -- Lecture 17 1110/10/2012
Note that the moment of inertia depends on both
a) The position of the rotational axisb) The direction of rotation
m m
d d
I=2md2
m m
d d
I=m(2d)2=4md2
PHY 113 A Fall 2012 -- Lecture 17 1210/10/2012
iclicker question:
Suppose each of the following objects each has the same total mass M and outer radius R and each is rotating counter-clockwise at an constant angular velocity of w=3 rad/s. Which object has the greater kinetic energy?
(a) (Solid disk) (b) (circular ring)
PHY 113 A Fall 2012 -- Lecture 17 1310/10/2012
Various moments of inertia:
solid cylinder:
I=1/2 MR2
solid sphere:
I=2/5 MR2
solid rod:
I=1/3 MR2
R
RR
PHY 113 A Fall 2012 -- Lecture 17 1410/10/2012
Calculation of moment of inertia:
Example -- moment of inertia of solid rod through an axis perpendicular rod and passing through center:
2222
31
22MRdrr
RM
rdrRM
rmIR
R
R
Riii
R
PHY 113 A Fall 2012 -- Lecture 17 1510/10/2012
Note that any solid object has 3 moments of inertia; some times two or more can be equal
j
i
k
iclicker exercise:Which moment of inertia is the smallest? (A) i (B) j (C) k
PHY 113 A Fall 2012 -- Lecture 17 1610/10/2012
iclicker exercise:Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?
AB C
2MRI A
22 5.02
1MRMRIB
22 4.05
2MRMRIC
PHY 113 A Fall 2012 -- Lecture 17 1710/10/2012
22
2
1
2
1
:object rolling
ofenergy kinetic Total
CM
CMrollingtotal
MvI
KKK
CMvRdt
dR
dt
dsdt
d
: thatNote
2
2
22
2
2
1
2
1
2
1
CM
CM
CMrollingtotal
vMR
I
MvRR
I
KKK
PHY 113 A Fall 2012 -- Lecture 17 1810/10/2012
iclicker exercise:Three round balls, each having a mass M and radius R, start from rest at the top of the incline. After they are released, they roll without slipping down the incline. Which ball will reach the bottom first?
AB C
2MRI A
22 5.02
1MRMRIB
22 4.05
2MRMRIC
