• 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

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

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

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

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

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

A wire is bent into a U-shape. Where is the CM?

i-Clicker

An L-shaped profile is cut from sheet metal.

Where is the CM?

i-Clicker

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

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

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?

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

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

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

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

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

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

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(

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

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

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

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

23

i-Clicker

rv

• 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!