Rotation and angular ;momentum; Examples; Explanation; Principle for a multiparticle system; Applications of Angular ;momentum;

1. Rotation and Angular
Momentum
CL-101 ENGINEERING MECHANICS
B. Tech Semester-I
Prof. Samirsinh P Parmar
Mail: samirddu@gmail.com
Asst. Professor, Department of Civil Engineering,
Faculty of Technology,
Dharmsinh Desai University, Nadiad-387001
Gujarat, INDIA

2. Content of the presentation
• Rotation and angular momentum
• Examples
• Explanation
• Principle for a multiparticle system
• Applications of Angular momentum

3. 3
Rotation and angular momentum
Consider an object moving in a circle.
 It has linear
momentum at
every point along
its trajectory:
p = mv
 But it also has angular momentum,
characteristic of its rotational motion.

4. 4
Rotation and angular momentum,
cont’d
• Conservation of Angular Momentum states that the total
angular momentum of an isolated system is constant.
• For our ball on a string:

5. 5
Rotation and angular momentum,
cont’d
• Conservation of momentum tells us:
• The right hand side must equal the left hand side.
• So as rf gets smaller, vf must get larger.

6. 6
Rotation and angular momentum,
cont’d
• In other words, if we shorten the string, the
ball’s speed increases.
• This is a
consequence
of conservation
of angular
momentum.

7. 7
Just as the ball’s speed increases as the
string’s length shortens, the boy spins faster
when he pulls in his arms.
Rotation and angular momentum,
cont’d

8. 8
A figure skater
spins faster when
she pulls in her
arms and legs –
and vice versa.
Rotation and angular momentum,
cont’d

9. 9
Rotation and angular momentum,
cont’d
• This also holds for orbits.
• As the satellite
gets to B, it
must move
faster than at
A.
mvArA = mvBrB

10. 10
Example
You spin a ball attached to the end of a 1.0-meter string with a
speed of 10 m/s. Find the ball’s speed as you shorten the string
to 10 centimeters (0.1 m).

11. Momentum vs. Angular Momentum
• Definition only valid for rigid bodies
(with well-defined angular velocity).
• Angular momentum is parallel to
angular velocity only if an object is
symmetric around the axis of rotation.

12. The angular momentum principle
dp
dt
= Fnet
dLA
dt
= ?
d rA
´ p
( )
dt
=
drA
dt
´ p + rA
´
dp
dt
= v ´g mv 0

= Fnet
The angular momentum principle
for a point particle
A
A
r net
F
p
dLA
dt
= rA
´ Fnet
DLA
= rA
´ Fnet
( )Dt
torque :
tA
º rA
´ Fnet
= tA
= tA
Dt
Note:
The angular momentum principle is derived from the momentum principle

13. Clicker
torque :
tA
º rA
´ Fnet
tA
= rA
Fnet
sinq
A

1. Which case corresponds to largest torque?
A
B
C
2. The direction of Torque in case B is:
A) into page
B) out of page
C) it has no direction since it is zero
A
r 
A A net
r F
 

Torque at angle

14. Example:
momentum and angular momentum principles
dp
dt
= Fnet
 
d mv
mg
dt

dv
g
dt

Use the momentum principle:
dLA
dt
= rA
´ Fnet
= tA
Use the angular momentum principle:
tA
tA
= xmg
A
L r p xmv

 
LA
= rA
´ p
LA
 
d xmv dv
xm
dt dt

dv
g
dt

Falling object
(nonrelativistic)

15. Conservation of angular momentum
DLA,system
+ DLA,surroundings
= 0
Example:
Important: both L’s must be about the same point (axis)

16. A comet
LA
= rA
´ p 1 1 2 2
rmv r mv
 1 1 2 2
rv r v

dLA
dt
= rA
´ Fnet
= tA
A
r
grav
F
CLICKER: What is the direction of the torque on the comet in point B about
the star due to gravitational pull?
A) Into the page
B) Out of the page
C) It is zero
B
(nonrelativistic)

17. Example: Kepler and elliptical orbits
Kepler, 1609: “a radius vector joining any planet to the Sun sweeps out
equal areas in equal lengths of time”
Can be easily proven using conservation of angular momentum
See book p. 430 (11.4)

18. Clicker
A ball falls straight down in the xy plane. Its momentum is shown by the
blue arrow. What is the direction of the ball's angular momentum
about the origin?
x
y
A) +y
B) –y
C) +z (out of the page)
D) –z (into the page)
E) zero magnitude
LA
= rA
´ p

19. Clicker
A planet orbits a star, in a circular orbit in the
xy plane. Its momentum is shown by the red
arrow.
What is the direction of the angular
momentum of the planet in respect to
the star?
A) same direction as
B) opposite to
C) into the page
D) out of the page
E) zero magnitude
p
p
p
LA
= rA
´ p

20. tnet,ext,A
Ltot,A
Multiparticle system
r1
r2
r3
F1,ext
f13
f12
f31
f23
f32
f21
d L1
+ L2
+ L3
( )
dt
=
dL1
dt
+
dL2
dt
+
dL3
dt
= r1
´ F1,ext
+ r2
´ F2,ext
+ r3
´ F3,ext
Net torque caused by internal forces cancels out!
m1 m2
m3
A
1 2
r r
 

F2,ext
F3,ext
dL1
dt
= r1
´ F1,ext
+ r1
´ f12
+ r1
´ f13
dL2
dt
= r2
´ F2,ext
+ r2
´ f21
+ r2
´ f23
dL3
dt
= r3
´ F3,ext
+ r3
´ f31
+ r3
´ f32

21. The angular momentum principle
for a multiparticle system
dLtot,A
dt
= tnet,ext,A
DLtot,A
= tnet,ext,A
Dt
The angular momentum principle relative to the center of mass:
dLcm
dt
=
d
dt
rcm,cm
´ Ptot
( )+ Lrot
é
ë
ù
û =
dLrot
dt
dLrot
dt
= tnet,cm
DLrot
= tnet,cm
Dt

22. The three principles of mechanics
Momentum Angular momentum Energy
dp
dt
= Fnet
dLA
dt
= tA
E W Q
  
External force:
momentum changes
External torque:
angular momentum
changes
Energy input:
energy of the system
changes
No external force:
momentum is constant
No external torque:
angular momentum
is constant
No energy input:
energy of the system
is constant
Location of object
does not matter
Location of object
relative to A
does matter
Location of object
does not matter
(fundamental) (fundamental)
(derived)

23. Angular momentum: a system with no torque
http://www.hep.phys.soton.ac.uk/courses/phys2006/
dLrot
dt
= tnet,cm
= 0
rot
L I

2 2
1 1 2 2 ...
I m r m r
 
  
i f
I I

f f i i
I I
 

i
f i
f
I
I
 

Dorothy
Hamill
,
1985

24. Angular momentum: a system with no torque
Cat always lands on its feet
http://www.youtube.com/watch?v=RHhXbOhK_hs

25. Angular momentum: application
A free-falling cat cannot alter its total
angular momentum. Nonetheless, by
swinging its tail and twisting its body to
alter its moment of inertia, the cat can
manage to alter its orientation
See also book example: High dive
page 437

26. Angular momentum: application
A meteor rips through
a satellite with solar panels.
Calculate:
vx,vy of center of mass
f – angular velocity
Momentum principle:
Mv + mv1
cosq
( ),mv1
sinq,0 = Mvx
+ mv2
cosq
( ), Mvy
+ mv2
sinq
( ),0
 
1 2 cos
x
m
v v v v
M

    
1 2 sin
y
m
v v v
M

 
x
y ω

27. Angular momentum: application
A meteor rips through
a satellite with solar panels.
Calculate:
vx,vy of center of mass
f – angular velocity
Angular momentum principle:
1 2
cos cos
i f
I mv h I mv h
   
  
 
1 2 cos
f i
hm
v v
I
  
  
For sphere:
2
2
5
I MR

Direction?
ω

28. Static equilibrium: seesaw
dLA
dt
= tnet,ext,A
= 0
tnet,ext,A
= r1
´ M1
g
( )+ r2
´ M2
g
( )= 0
M1
r1
+ M2
r2
( )´ g = 0
-M1
r1
= M2
r2
1 1 2 2
,0,0 ,0,0
M d M d
  
1 1 2 2
M d M d

A
1
r 2
r
rcm
=
M1
r1
+ M2
r2
M1
+ M2
= 0