Download Physics 7B - AB Lecture 7 May 15 Angular Momentum Model

Physics 7B - AB
Lecture 7
May 15
Recap Angular Momentum Model
(Second half of Chap 7)
Recap Torque, Angular Momentum
Rotational Inertia (new concept!)
Intro to Newtonian Model
(start Chapter 8)
1
Re-evaluation Request Due
Quiz 2 May 22 (next Thursday)
Quiz 3 May 29
Quiz 3 average 7.75 (C+)
Quiz 4 graded, solution up on the
web, rubrics will follow
2
Recap Rotational (Angular) Motion
1. A ladybug sits at the outer edge of a merry-go-round, and a
gentleman bug sits halfway between her and the axis of
rotation. The merry-go-round makes a complete revolution
once each second. The gentleman bug’s angular speed,
, is is
gentleman
bug
z
A) half the ladybug’s.
B) the same as the ladybug’s.
C) twice the ladybug’s.
D) impossible to determine.
y
x
Gentleman
bug
Ladybug
3
Recap Rotational (Angular) Motion
1. A ladybug sits at the outer edge of a merry-go-round, and a
gentleman bug sits halfway between her and the axis of
rotation. The merry-go-round makes a complete revolution
once each second. The gentleman bug’s angular speed,
, is is
gentleman
bug
z
A) half the ladybug’s.
B) the same as the ladybug’s.
C) twice the ladybug’s.
D) impossible to determine.
y
x
Gentleman
bug
Ladybug
B) The two bugs each travel the same angle
(ie. one revolution or 2p radians or any
other angle) in the same amount of time
so they have the same angular velocity
4
Recap Rotational (Angular) Motion
2. The ladybug’s angular velocity vector,
, points
along
 ladybug
points along
the the
A)
B)
C)
D)
E)
+
+
+
x-axis
x-axis
y-axis
y-axis
z-axis
z
y
x
Ladybug
5
Recap Rotational (Angular) Motion
2. The ladybug’s angular velocity vector,
, points
along
 ladybug
points along
the the
A)
B)
C)
D)
E)
+
+
+
x-axis
x-axis
y-axis
y-axis
z-axis
z
y
x
Ladybug
E) Use your right hand to show that the
angular velocity points along the +z
axis.
6
Recap Rotational (Angular) Motion
3. The wheel shown to the right can rotate
freely about its axle (the dot in the center)
which is fixed in space like the wheel in
lecture. A force, F, is applied as shown
F
(along with whatever other forces are acting
on it). F produces a torque aboutthethe
of mass.
pivotcenter
point.
is
the
direction
of torque
this torque?
What
is the
direction of this
?
What
F
A) Clockwise
away rotation)
from you).
A)
Torque vector(torque
 pointingvector
into thepointing
slide (clockwise
B) Counterclockwise (torque vector pointing toward you).
B)
Torque vector
 pointing out of the slide (counterclockwise rotation)
C) Torque
is zero.
C) Torque is zero.
7
Recap Rotational (Angular) Motion
3. The wheel shown to the right can rotate
freely about its axle (the dot in the center)
which is fixed in space like the wheel in
lecture. A force, F, is applied as shown
F
(along with whatever other forces are acting
on it). F produces a torque aboutthethe
of mass.
pivotcenter
point.
is
the
direction
of torque
this torque?
What
is the
direction of this
?
What
F
A) Clockwise
away rotation)
from you).
A)
Torque vector(torque
 pointingvector
into thepointing
slide (clockwise
B) Counterclockwise (torque vector pointing toward you).
B)
Torque vector
 pointing out of the slide (counterclockwise rotation)
C) Torque
is zero.
C) Torque is zero.
B) You should see that pushing this way
will tend to rotate the wheel
counterclockwise about its axle and that
the right hand rule give you a torque
vector pointing toward you.
8
Recap Rotational (Angular) Motion
4. The picture to the right shows the
vtablecloth
famous “yank the tablecloth out from
vcloth
under the wine goblet” trick. The
tablecloth is pulled to the right as shown. The forces on
the goblet are shown below the picture. The friction force
by the tablecloth produces a torque about the center of mass.
What direction is this torque?
A) Torque vector due to
friction force points
Fgravity by earth
away from you.
FEarth on
the goblet
on goblet
B) Torque vector due to
Ffriction
F
/ /tablebycloth
on the goblet
tablecloth
friction force points
on goblet
Fnormal by tablecloth
toward you.
(friction)
F table
cloth on the goblet
on
goblet
C) Torque is zero.
9
Recap Rotational (Angular) Motion
4. The picture to the right shows the
vtablecloth
famous “yank the tablecloth out from
vcloth
under the wine goblet” trick. The
tablecloth is pulled to the right as shown. The forces on
the goblet are shown below the picture. The friction force
by the tablecloth produces a torque about the center of mass.
What direction is this torque?
A) Torque vector due to
friction force points
Fgravity by earth
away from you.
FEarth on
the goblet
on goblet
B) Torque vector due to
Ffriction
F
/ /tablebycloth
on the goblet
tablecloth
friction force points
on goblet
Fnormal by tablecloth
toward you.
(friction)
F table
cloth on the goblet
on
goblet
C) Torque is zero.
B) Very similar to the previous question.
If the goblet tips over, it will tip
over counterclockwise.
10
Torque - rotational force that can
change the rotational motion
Force is exerted tangentially on the rim, the rim is
at a distance r (moment arm) from the pivot point.
Ftangential
r
Direction of Torque
Force and Torque are
two different physical quantities!
Direction of 
is given by the RHR
Torque this way
Rotation this way
11
Rotational motion is changed by applying forces, But
where the force is applied is Just as important as the
size of the force
Ftangential
r
Magnitude of Torque
 = r Ftangential = r Ftangential
12
Force and Torque are two different physical quantities!
Recap Extended Force Diagram
L/4
5. The uniform plank shown to
the right has a length, L,
and a mass M.
The plank
rests on a fulcrum and a
chunk of metal with mass M/2
hangs by a string from the left end.
downward force, F
Fby string on plank, be placed?
Where should the
string on Plank
A)
B)
C)
D)
E)
At the left end where
L/4 from the left end
L/2 from the left end
At the right end.
The force is equal to
the string is tied.
where the fulcrum is.
where the center of mass is.
zero so it doesn’t matter.
13
Recap Extended Force Diagram
L/4
5. The uniform plank shown to
the right has a length, L,
and a mass M.
The plank
rests on a fulcrum and a
chunk of metal with mass M/2
hangs by a string from the left end.
downward force, F
Fby string on plank, be placed?
Where should the
string on Plank
A)
B)
C)
D)
E)
At the left end where
L/4 from the left end
L/2 from the left end
At the right end.
The force is equal to
the string is tied.
where the fulcrum is.
where the center of mass is.
zero so it doesn’t matter.
14
Recap Extended Force Diagram
L/4
6. The uniform plank shown to
the right has a length, L,
and a mass M.
The plank
rests on a fulcrum and a
chunk of metal with mass M/2
hangs by a string from the left end. Where should the upward
force,F Fby fulcrum on plank, be placed?
fulcrum on Plank
A)
B)
C)
D)
E)
At the left end where
L/4 from the left end
L/2 from the left end
At the right end.
The force is equal to
the string is tied.
where the fulcrum is.
where the center of mass is.
zero so it doesn’t matter.
15
Recap Extended Force Diagram
L/4
6. The uniform plank shown to
the right has a length, L,
and a mass M.
The plank
rests on a fulcrum and a
chunk of metal with mass M/2
hangs by a string from the left end. Where should the upward
force, FFby fulcrum on plank, be placed?
fulcrum on Plank
A)
B)
C)
D)
E)
At the left end where
L/4 from the left end
L/2 from the left end
At the right end.
The force is equal to
the string is tied.
where the fulcrum is.
where the center of mass is.
zero so it doesn’t matter.
16
Recap Extended Force Diagram
L/4
7. The uniform plank shown to
the right has a length, L,
and a mass M.
The plank
rests on a fulcrum and a
chunk of metal with mass M/2
hangs by a string from the left end.
downward force, FFby earth on plank, be placed?
Where should the
Earth on Plank
A)
B)
C)
D)
E)
At the left end where
L/4 from the left end
L/2 from the left end
At the right end.
The force is equal to
the string is tied.
where the fulcrum is.
where the center of mass is.
zero so it doesn’t matter.
17
Recap Extended Force Diagram
L/4
7. The uniform plank shown to
the right has a length, L,
and a mass M.
The plank
rests on a fulcrum and a
chunk of metal with mass M/2
hangs by a string from the left end.
downward force, FFby earth on plank, be placed?
Where should the
Earth on Plank
A)
B)
C)
D)
E)
At the left end where
L/4 from the left end
L/2 from the left end
At the right end.
The force is equal to
the string is tied.
where the fulcrum is.
where the center of mass is.
zero so it doesn’t matter.
18
Recap Rotational (Angular) Motion
L/4
8. The uniform plank shown to the
right has a length, L, and a mass
The weight hanging from the
M.
left end of the plank has a mass
In order to
equal to M/2.
balance the plank as shown, a force has been exerted on the right end
of the plank. How large is this force and what direction is it in?
A) The force is
upwards.
B) The force is
upwards.
C) The force is
downwards.
D) The force is
downwards.
E) The force is
bigger than the weight on the left and it is exerted
smaller than the weight on the left and it is exerted
bigger than the weight on the left and it is exerted
smaller than the weight on the left and it is exerted
equal to zero.
19
Ffulcrum on Plank
F ??
L/4
8. The uniform plank shown to the
right has a length, L, and a mass
The weight hanging from the
M.
left end of the plank has a mass
In order to
equal to M/2.
the
has been exerted
force
balance the plank as shown,
Fstring onaPlank
= – (M/2)g
FEarth ononPlank
= –right
Mg end
of the plank. How large is this force and what direction is it in?
A) The force is
upwards.
B) The force is
upwards.
C) The force is
downwards.
D) The force is
downwards.
E) The force is
bigger than the weight on the left and it is exerted
smaller than the weight on the left and it is exerted
bigger than the weight on the left and it is exerted
smaller than the weight on the left and it is exerted
equal to zero.
20
Angular analogue to Impulse is
Angular Impulse
• Angular Impulse Is related to the net external
torque in the following way:
Net Angular Impulseext = ∆ L = ∫ ext(t)dt
If the torque is constant during a time interval ∆t
Net Angular Impulseext = ∆ L =  ave.ext x ∆ t
If the net torque is zero, the plank will stay stationary…
21
Ffulcrum on Plank
F
L/4
8. The uniform plank shown to the
right has a length, L, and a mass
The weight hanging from the
M.
left end of the plank has a mass
In order to
equal to M/2.
the
has been exerted
force
balance the plank as shown,
Fstring onaPlank
= – (M/2)g
FEarth ononPlank
= –right
Mg end
of the plank. How large is this force and what direction is it in?
A) The force is
upwards.
B) The force is
upwards.
C) The force is
downwards.
D) The force is
downwards.
E) The force is
bigger than the weight on the left and it is exerted
smaller than the weight on the left and it is exerted
bigger than the weight on the left and it is exerted
smaller than the weight on the left and it is exerted
equal to zero.
22
Ffulcrum on Plank
F = (M/6)g
L/4
8. The uniform plank shown to the
right has a length, L, and a mass
The weight hanging from the
M.
left end of the plank has a mass
In order to
equal to M/2.
the
has been exerted
force
balance the plank as shown,
Fstring onaPlank
= – (M/2)g
FEarth ononPlank
= –right
Mg end
of the plank. How large is this force and what direction is it in?
A) The force is
upwards.
B) The force is
upwards.
C) The force is
downwards.
D) The force is
downwards.
E) The force is
bigger than the weight on the left and it is exerted
smaller than the weight on the left and it is exerted
bigger than the weight on the left and it is exerted
smaller than the weight on the left and it is exerted
equal to zero.
 ave.ext = (L/4)(M/2)g + (0)(M/2)g – (L/4)Mg + (3L/4) F = 0
(L/4)(M/2)g – (L/4)Mg + (3L/4) F = 0
(3L/4) F = (L/4)(M/2)g
3 F = (M/2)g
F = (M/6)g
23
Define Angular Momentum L
Think of rotational inertia I kind of like mass for now.
rotating like this with 
Magnitude of Angular Momentum
L=I
24
Define Angular Momentum L
Think of rotational inertia I kind of like mass for now.
Direction of L
is given by the RHR
rotating like this with 
Magnitude of Angular Momentum
L=I
Rotation
This way
25
What do you mean by rotational analog to mass?
Rotational Inertia
Depends not only on the amount of mass in the object but also on how
the mass is distributed about the axis of rotation : I can change!
Formula not really important, but the idea is that the further mass is from
the axis of rotation, the greater I
Example: Same mass, same volume but arranged
differently
r2
r1
rotates this way
I1
rotates this way
>
I2
26
What do you mean by rotational analog to mass?
Rotational Inertia
the idea is that the further mass is from the axis of rotation, the greater I
Point mass m
r
Thin ring of mass m
r
Disk of mass m
r
27
What do you mean by rotational analog to mass?
Rotational Inertia
the idea is that the further mass is from the axis of rotation, the greater I
Point mass m
r
Thin ring of mass m
r
Disk of mass m
r
I = mr2
28
What do you mean by rotational analog to mass?
Rotational Inertia
the idea is that the further mass is from the axis of rotation, the greater I
Point mass m
r
I = mr2
Thin ring of mass m
r
Disk of mass m
r
I = mr2
29
What do you mean by rotational analog to mass?
Rotational Inertia
the idea is that the further mass is from the axis of rotation, the greater I
Point mass m
r
I = mr2
Thin ring of mass m
r
I = mr2
Disk of mass m
r
I = (1/2)mr2
30
Question
31
Question
32
Why does a figure skater start spinning
faster when she pulls her arms in?
Initial
initial
Final
<
final
33
Assume ice surface is frictionless, the net
torque on the skater is zero…
Net Angular Impulseext = ∆ L =  ave.ext x ∆ t = 0
Angular Momentum Lskater is conserved!
Initial L initial, skater = I initial, skater initial
Final L final, skater = I final, skater final
I initial, skater initial = I final, skater final
Remember I initial, skater > I final, skater
So when the rotational inertia decreases (which it does when
she pulls her arms in), angular velocity must increase in
order to conserve the angular momentum
Spin control is nothing but invoking Conservation of
Angular momentum!
34
Newtonian Model
Ouch…
Umm why does an apple fall ??
I tried to understand the force on
an apple and its relation to
apple’s motion. It is all
summarized in Newton’s Laws of
Motion.
Sir. Isaac Newton
35
Newtonian Model
Newton’s Laws of Motion
Newton’s first law:
If the momentum changes, there is a net force on the system.
If the momentum is not changing, there is no net force on the
system.
Net Impulseext = ∆ p = ∑ Fave.ext x ∆ t
Newton’s second law:
Quantitatively relates instantaneous change in momentum (or
velocity) to net force
( ∑ Fave.ext = ∆ p /∆ t )
in terms of instantaneous time rate change of momentum…
∑ Fext = d p /dt = m dv/dt = ma
An unbalanced force (∑ Fext  0) causes a change in motion of an
36
object,i.e.time rate change of velocity (acceleration)
Newtonian Model
Newton’s Laws of Motion
Newton’s first law:
If the momentum changes, there is a net force on the system.
If the momentum is not changing, there is no net force on the
system.
Net Impulseext = ∆ p = ∑ Fave.ext x ∆ t
Newton’s second law:
Quantitatively relates instantaneous change in momentum (or
velocity) to net force
∑ Fext = d p /dt = m dv/dt = ma
An unbalanced force (∑ Fext  0) causes a change in motion of an
object,i.e.time rate change of velocity (acceleration)
Newton’s third law: F A on B = – F B on A
You cannot push without being pushed yourself!
37
When is Newtonian Model valid ?
When things are not too small
and
its motion not too fast
Protons
+ Neutrons
Quic kTime™ and a
TIFF (Unc ompres sed) decompress or
are needed to see this picture.
Electrons
Air France Concorde
Mach 2.23 = 7.58 x 102 m/s << Speed of light (3 x 108 m/s)
38
How is Newtonian Model useful ?
We know the net force on the object and want to
know what its subsequent motion is
OR
We know the motion of an object and want to know
details of the forces acting on the object
We know aconcorde, What is F ejected gas on the concorde?
We know the gravitational forces between all the planets39and
the sun. When/Where is next total solar eclipse?
Practice Newton’s Laws of Motion
1.
A) for True or B) for False. A box rests on the floor.
The upward force exerted by the floor surface on the box
forms a Newton’s 3rd Law pair with the force exerted
downward on the box by the Earth’s gravity.
B) False!
According to Newton’s 3rd Law:
Fgravity by earth on box = -Fgravity by box on earth
And
Fnormal by floor on box = -Fnormal by box on floor
40
Practice Newton’s Laws of Motion
1.
A) for True or B) for False. A box rests on the floor.
The upward force exerted by the floor surface on the box
forms a Newton’s 3rd Law pair with the force exerted
downward on the box by the Earth’s gravity.
B) False!
According to Newton’s 3rd Law:
Fgravity by earth on box = -Fgravity by box on earth
And
Fnormal by floor on box = -Fnormal by box on floor
41
Practice Newton’s Laws of Motion
2.
A child sits on a stationary swing held up by chains. The
child feels the normal force up from the seat of the swing.
Which force is the equal-and-opposite pair to this one,
according to Newton's 3rd Law?
A)
B)
C)
D)
E)
Force of child's weight down on chain.
Tension force up on seat by chain.
Gravity force down on child.
Normal force down on seat by child.
none of the above.
D)
According to Newton’s 3rd Law:
Fnormal by seat on child = -Fnormal by child on seat
42
Practice Newton’s Laws of Motion
2.
A child sits on a stationary swing held up by chains. The
child feels the normal force up from the seat of the swing.
Which force is the equal-and-opposite pair to this one,
according to Newton's 3rd Law?
A)
B)
C)
D)
E)
Force of child's weight down on chain.
Tension force up on seat by chain.
Gravity force down on child.
Normal force down on seat by child.
none of the above.
D)
According to Newton’s 3rd Law:
Fnormal by seat on child = -Fnormal by child on seat
43
Position vs Velocity vs Acceleration
3. The graph to the right shows the
position as a function of time for
two trains running on parallel
tracks. Which is true?
B
A) At time, tB, both trains have the
same velocity.
B) Both trains speed up all the time.
C) Both trains have the same velocity
at some time before tB.
D) None of the above are true.
C) velocity is
slope of position
graph so both
trains have the
same velocity at
time t1
A
position
tB
t1
time
44
Position vs Velocity vs Acceleration
3. The graph to the right shows the
position as a function of time for
two trains running on parallel
tracks. Which is true?
B
A) At time, tB, both trains have the
same velocity.
B) Both trains speed up all the time.
C) Both trains have the same velocity
at some time before tB.
D) None of the above are true.
C) velocity is
slope of position
graph so both
trains have the
same velocity at
time t1
A
position
tB
t1
time
45
Be sure to write your name, ID number & DL section!!!!!
1
MR 10:30-12:50
Dan Phillips
2
TR 2:10-4:30
Abby Shockley
3
TR 4:40-7:00
John Mahoney
4
TR 7:10-9:30
Ryan James
5
TF 8:00-10:20
Ryan James
6
TF 10:30-12:50
John Mahoney
7
W 10:30-12:50
Brandon Bozek
7
F 2:10-4:30
Brandon Bozek
8
MW 8:00-10:20
Brandon Bozek
9
MW 2:10-4:30
Chris Miller
10 MW 4:40-7:00
Marshall Van Zijll
11 MW 7:10-9:30
Marshall Van Zijll
46