Transcript Angular Momentum (print version)

Angular Momentum (l) -
 
l rp


l  r  mv
l  rmv sin 
Units:
m
m kg  m s  kgm2
s
Angular Momentum is a vector whose direction is
perpendicular to the plane containing r and p given by the
right hand rule.
1. A particle P with mass 2.0 kg has a position vector


r r  3.0 m  and velocity v v  4.0 m as shown.
s

It is acted upon by a force F F  2.0 N . All three
vectors lie in the xy plane. About the origin, what are


F


v
30.
P

r
a. the angular momentum of the particle,


l  r  mv
l  rmv sin 
30.
45


l  3.0 m 2.0 kg  4.0 m sin 150
s
2
kgm
Out of the page
l  12
s
 kˆ 
150

v
30.

r
1. A particle P with mass 2.0 kg has a position vector


r r  3.0 m  and velocity v v  4.0 m as shown.
s

It is acted upon by a force F F  2.0 N . All three
vectors lie in the xy plane. About the origin, what are


F


v
30.

r
b. and the torque acting on the particle.
 
  r F
  rF sin 
45
  3.0 m 2.0 N sin 30.
  3.0 Nm
Out of the page
 kˆ
 

F
30.

r
30.
P
2. Two objects are moving as shown.
What is their total angular momentum
about point O?

r
l


since r  mv
6.5 kg
1
s
1.5 m
is into the page
O

v
2 .2 m
3 .6 m
2.8 m
L  l2  l1

v
l


since r  mv
is out of the page

r
2
3.1 kg
s
2. Two objects are moving as shown.
What is their total angular momentum
about point O?
6.5 kg
1
2 .2 m
s
1.5 m
O
3 .6 m
2.8 m
2
3.1 kg
L  l2  l1
L  r2 m2v2 sin 90.  r1m1v1 sin 90.

L  2.8m 3.1kg  3.6 m
2
kgm
L  9.8
s
s  1.5m6.5kg 2.2 m s 
Out of the page
 kˆ
 
s
y
3. What is the angular momentum of a
rigid object rotating about a fixed axis?


li  ri  mi vi
li  ri mi vi sin 90.

ri
li  ri mi vi
n
L   ri mi vi
i 1
But
vi  ri
n
L   ri mi ri 
i 1
z
mi

vi
x
y
3. What is the angular momentum of a
rigid object rotating about a fixed axis?
n
L   ri mi ri 
L
i 1
But
  constant for all mi
L    mi ri2 
 i 1

n
But

ri
z
n
I   mi ri2
i 1
L  I
Angular Momentum of a rigid
object rotating about a fixed axis
mi

vi
x
3. Three particles, each of mass m, are
fastened to each other and to a rotation
axis by three massless strings, each with
length l. The combination rotates around
the rotational axis at O with angular velocity
ω in such a way that the particles remain in
a straight line. In terms of m, l and ω and
relative to point O, what are
a. the rotational inertia of the combination,

m
l
m
l
O
l
3
I   mi ri2
i 1
I  ml 2  m2l 2  m3l 2
I  ml 2  4ml 2  9ml 2
I  14ml 2
m
3. Three particles, each of mass m, are
fastened to each other and to a rotation
axis by three massless strings, each with
length l. The combination rotates around
the rotational axis at O with angular velocity
ω in such a way that the particles remain in
a straight line. In terms of m, l and ω and
relative to point O, what are
O
b. the angular momentum of the middle
particle,
2 methods

m
l
m
l
l
Treat as a separate object


lm  r  mv
lm  rmv sin 
lm  2l mv sin 90
lm  2lmv
m
But
v  r
v   2l 
lm  2lm2l 
lm  4ml 2
3. Three particles, each of mass m, are
fastened to each other and to a rotation
axis by three massless strings, each with
length l. The combination rotates around
the rotational axis at O with angular velocity
ω in such a way that the particles remain in
a straight line. In terms of m, l and ω and
relative to point O, what are
O
b. the angular momentum of the middle
particle,
2 methods
Treat as a rigid object
lm  I m
But
I m  4ml 2
lm  4ml 2

m
m
l
m
l
l
3. Three particles, each of mass m, are
fastened to each other and to a rotation
axis by three massless strings, each with
length l. The combination rotates around
the rotational axis at O with angular velocity
ω in such a way that the particles remain in
a straight line. In terms of m, l and ω and
relative to point O, what are
c. the total angular momentum of the
three particles.

m
m
l
m
l
O
L  I
L  14ml 2
l