Page 1
• No lecture, no quiz, no minilabs, no homework next
week.
• But: Mandatory Post-Test!
• Pre-test, pre-survey, post-test, post-survey count 2
points toward your course grade.
• Mo, Thu, Fri sections take post-test during recitation on
Mo, Tu, We next week.
• Tu and We sections take post-test during last week of
classes (Tu Dec. 9 and We Dec. 10).
• Make-up sessions for post-test:
1. During lecture periods on We next week.
2. On Mo Dec. 8.
Diagnostic Post Test
Page 2
CENTER OF MASS MOTION & ROTATIONAL KINEMATICS
REVIEW OF LAST LECTURE:
• MOMENTUM:
• IMPULSE OF A FORCE:
(Strong force acting for short time)
• Impulse-Momentum Theorem
• CONSERVATION OF MOMENTUM:
• TYPES OF COLLISIONS:
• MOMENTUM CONSERVED IN COLLISIONS
(When collision force >> external force)
Elastic: conserved
Inelastic: only is conserved
Completely Inelastic: only conserved & objects stick
vmp
dt
pdamF
)( 12 ttFJ
2
1
t
tdtFJ
12 ppJ
then 0 If IFext PPF
][][FFII BABA pppp
KEP &
P
P
t1 t2 t
xF
xF
Page 3
Many nuclear reactors use Li (M = 7mo) to reduce the KE of
neutrons (M n = mo) produced
in nuclear reactions by elastic
collisions with Li nuclei.. An
alternative is Na (M Na = 23
mo).
Li is a better choice because:
A. Ptot is lower after n Li collision
B. n has less KE after n Li collision compared
to n Na collision
C. Ptot is higher after n Na collision
D. KE of n does not change in an elastic collision
E. Na is better choice (trick question)
i-Clicker
Li n
Na n
- or -
Pfinal = Pinitial
Page 4
CENTER OF MASS
Point in an extended mechanical system that
moves as though all the mass were concentrated
at that point
Consider a collection
of (different) masses
distributed on x-axis.
Define
center of mass: ...
...
321
332211
mmm
xmxmxmxcm
i.e.:
Similar expression for and
If mass is a continuous distribution:
cmycmz
M
xm
m
xmx
ii
i
ii
cmi
i
i
M
rmr
ii
cmi
dmrrMcm
1
Page 5
EXAMPLE:
mm
mmxcm
3
)2(3)4(
The center of mass is the mass-weighted
average position of the particles.
2
1
4
2
4
))6(4(
x
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LOCATING CM
• If body is homogeneous and has a geometric center
(e.g., uniform sphere or cube) then CM is geometric
center
• CM of symmetric body is along axis of symmetry
(cylinder, wheel, dumbbell)
• For collection of extended bodies, find CM of each
body, then CM of collection is CM of those point
masses
EXAMPLE: Two meter sticks, each with a mass of
0.25 kg, form a “T”. Where is the CM?
m 25.0)kg 5.0(
)m 5.0)(kg 25.0(0)kg 25.0(cm
y
0cm x
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A wire is bent into a U-shape. Where is the CM?
i-Clicker
Page 8
An L-shaped profile is cut from sheet metal.
Where is the CM?
i-Clicker
Page 9
The right half of a meter stick has twice the density
of the left half. Where is the CM?
i-Clicker
A B C E D
Page 10
MOTION OF CM
• Under influence of external force CM moves like
an imaginary particle of mass M.
• Total momentum of collection of masses can only
be changed by an external force. P
M
rmr iii
cm
i
iicm rmrM
i i
iiicm PpvmvMdt
d
extcm F
dt
PdaM
dt
d
again
dt
PdaMF cmext
Page 11
Mass m moves along x-axis with speed v
towards mass 3m which is at rest.
They collide and stick.
What is the motion of the CM before and
after the collision?
Before
After
No External Forces:
After the collision CM moves with the combined mass:
Before collision:
IF PP
33 i
fif
vvmvmv
3v
cmv
3
2
3
2 2121 xx
m
mxmxxcm
EXAMPLE:
33
)0(2
3
2 21
cmcm
iidt
dx
dt
dxvv
dt
dxv
Why does the velocity of the CM not change?
Page 12
A rocket ship moves in a gravity-free region of space
with constant velocity. It fires a short burst of gas
from the rear engine. Afterwards, the CM of the
rocket and gas system has:
A) Sped up
B) Slowed down
C) Has same constant velocity
D) Changed but can’t tell how
E) Insufficient info to tell anything
i-Clicker
cmext aMF
00 cmext aMF
Page 13
ROCKET PROPULSION
Conservation of momentum of rocket and propellant !!
No external forces:
Rocket is propelled by recoil !
fi PP
exf
exf
exf
mvvvM
mvMvMv
vvmMvvmM
)(
)()(
M + m
m
Page 14
RIGID BODY ROTATION
• All points rotate about same
axis (through 0, to page)
• If we follow motion of point P as
body rotates, only q changes
Use polar coordinates
Measure q in radians (fractions of 2p )
arc length: (dimensionless) r
srs qq
rad 22
360 pp
r
r
3.572
360rad 1
p
deg][ 180
[rad] :or qp
q
ANGULAR VELOCITY )( z
Rate of change in angle (in xy plane)
Average Angular Velocity:
Instantaneous:
about z - axis
tttavz
qqq
12
12
dt
d
ttz
qq
lim0
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zcw)(ccw)(
ANGULAR ACCELERATION )( z
• Average:
• Instantaneous:
Rigid Body Rotation: Every particle has the same
Define vectors:
• Point along axis of rotation
• Have magnitude of and • Orientation define by Right Hand Rule
Sign of : (+) ccw
(-) cw
Sign of :
Units of : s
rad 2
s
rev1 p
dt
d from
zz and
ttt
zz
zav
12
12
2
2
dt
d
dt
dz
q
and
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RIGID BODY ROTATION WITH CONSTANT
ANGULAR ACCELERATION
Form of equations same as 1-D motion:
Leads to analogous Equations of Motion:
2
2
)(
dt
xd
dt
dva
dt
dxv
tx
xx
x
1D: Fixed Axis Rotation
tvvxx
xxavv
tatvxx
tavv
a
oxxo
oxox
xoo
xox
x
x
x
x
)(
)(2
const
21
22
2
21
2
2
)(
dt
d
dt
ddt
d
t
zz
z
q
q
q
t
tt
t
ozzo
ozozz
zozo
zozz
z
)(
)(2
const
21
22
2
21
qq
qq
qq
Page 17
RELATING LINEAR AND ANGULAR MOTION
Each point, P, moves in circle
Magnitude of
Total acceleration
dt
dr
dt
dsv
q
ta
rdt
dr
dt
dvat
ra
22
rr
var
rt aaa
rv
Magnitude of centripetal acceleration
Page 18
EXAMPLE: A bike tire is rotated by 120o to get the stem
vertical. By how many radians was the tire rotated?
EXAMPLE: Your bike tires have a diameter of 0.75 m.
You ride from CAC BC at a speed of 18 km/hr.
What is the angular velocity of your tires?
• in rev/s?
• in RPM?
3
2)120(
180rad
pp
m/s 5s/hr 106.3
m/hr 10183
3
v
r
v
rad/s 3.13m 325.0
m/s 5
rev/s 12.2rad/rev 2
rad/s 3.13
p
rpm 127s/min) 06rev/s)( 12.2(
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EXAMPLE (Cont.): Suppose it took you 4 sec to reach
the speed of 18 km/hr. What was the (constant)
angular acceleration of your tires during this time?
• How many REVs did the tires make while accelerating?
to
sec) 4(0rad/s) 3.13(
2rad/s 33.3
2
21 ttoo qq
22
21 s) 4)(rad/s 33.3(q
rad 6.26q
rev 24.4rad/rev 2
rad 6.26
pq
Page 20
The graphs below show the angular velocity of two
objects during the same time interval. What is true
about the objects final angular displacement?
A. Object 1 has a greater angular displacement.
B. Object 2 has a greater angular displacement.
C. Object 1 and Object 2 have the same final angular
displacement.
D. It cannot be found with the information given.
i-Clicker
Page 21
Sketch an angular velocity versus time graph given
the angular acceleration graph shown for the same
time interval, assuming the initial angular velocity is
zero.
i-Clicker
Α
C
Β
D
Page 22
Sketch an angular acceleration versus time graph given the
angular velocity versus time graph shown for the same
time interval.
Α
C
Β
D
Ε
i-Clicker
Page 24
• No lecture, no quiz, no minilabs, no homework next
week.
• But: Mandatory Post-Test!
• Pre-test, pre-survey, post-test, post-survey count 2
points toward your course grade.
• Mo, Thu, Fri sections take post-test during recitation on
Mo, Tu, We next week.
• Tu and We sections take post-test during last week of
classes (Tu Dec. 9 and We Dec. 10).
• Make-up sessions for post-test:
1. During lecture periods on We next week.
2. On Mo Dec. 8.
Diagnostic Post Test
Happy Thanksgiving!