Discussion: Momentum Principle
Dp = Fnet Dt is the “discrete form”; there is also a “continuous form” where the deltas
become so small, they get “dinky.”
Ask for students to list (3) ways left and right sides are different. Similar. Call on
random tables; discuss.
Link to interactions.
Emphasize that this is the first fundamental principle; not a definition (it actually tells us
something).
Discussion: Momentum Update
Recall the Position Update equation: rfinal = rinitial +vt. In your groups (and on
whiteboards), I’d like you to predict a “momentum update” equation that has the same
form. Notice that this is the same as the momentum principle, just in a useable format.
Note an example where delta(p) is the same for each step, but p isn’t. Any constant force
example – a ball falling, for example.
Tangible: Don’t flip me off
WID 1066802 movies
Flipbook.doc
Fan backward.MOV
Fan forward.MOV
Fan off.MOV
Now that you’ve seen one case and done an example, we’re going to look at several.
On YouTube, there are three movies of cart rolling on a track.
Recorder (Fan forward), Skeptic (Fan Backwards)
Manager (Fan off),
1) Watch the video below that corresponds to your role:
Manager
Recorder
Skeptic
(Note that the markers in front of the track are 10 cm apart.)
2) After you watch your video, write a detailed description of what you just saw. Pay
attention to how the cart moved.
3) Pass your detailed description to one of your group members (Manager pass to recorder,
recorder to skeptic, skeptic to manager).
Chapter 2
1
4) Based on the description you read, draw a flipbook movie of the described motion on the
flipbook passed out in class.
5) Now, pass the description and flipbook you created to the remaining group member
(Manager pass to recorder, recorder to skeptic, skeptic to manager).
6) Now you have a description and a flip book to go with it. WATCH the fan cart movie that
the description was based on. How do the description and flip book compare to the video?
Discuss: What should the flip books look like for each case?
We’ll do graphs later.
Discuss: relate to momentum principle and position update formula!
Discussion: How to read the textbook
[Note: this activity has traditionally been one of the highest-rated activities in the start of the semester.
Students appreciate seeing a careful look at how reading the textbook is different from reading a novel,
and they report that this helps them. During the discussions, I'd always feel like they weren't really
caring (paying attention elsewhere) but they seem to think it's worthwhile.]
Example on page 76…to see how to read the book!
Read title of example, extract physics quantities: m = 0.5 kg, ri = <0,0,0> m, vi = <3,7,0> m/s.
For each part (a) and (b) extract what you are trying to find.
Relate procedure to other examples.
Equation comes from pg 73
Think about meaning of two possible solutions for time.
Chapter 2
2
Chapter 2
3
Ponderable: Activity - Predicting the future
WID 1066810
crystalball
How we do “problems” in physics class: you need to start with a physics principle. Then
decide how you can utilize that principle to answer the problem. Let’s talk about how
we’re going to start this problem, then you do it.
Fan cart mass 400g, constant net force due to air and friction is <0.20, 0, 0> N. Release
from rest at 0.15 m. Where is it 1.0 s later?
System: cart w/fan. Indicated by a circle as shown here
Surroundings: Earth, track, air
Initial time = 0 Final time = 1.0 s
Momentum principle
The class then does the following, in groups:
(
)
pf = pi + Fnet Dt = pi + Ftrack + FEarth + Fair Dt
Ftrack = -FEarth
pf = 0 + 0.20, 0, 0 N (1.0 s ) = 0.20, 0, 0 N × s
0.20, 0, 0 kg×m
×s
pf
s2
vf =
=
= 0.50, 0, 0
m
0.400 kg
rf = ri + vavg Dt = 0.15, 0, 0 m + 0.25, 0, 0
m
s
m
s
vi = 0, 0, 0
(1.0 s ) =
m
s
vavg = 0.25, 0, 0
m
s
0.40, 0, 0 m
Make sure you point out the position update formula and the fact that the average
velocity comes from the very important assumption that the net force is CONSTANT
over the time interval that we're looking at.
Check: Speed increased, which is reasonable since force in same direction as momentum.
Note: This is correct calculation of average velocity only when net force is constant.
See example pg 55, where initial velocity is <0.3,0,0> m/s and final time is 0.6 s.
Chapter 2
4
Ponderable: Activity - You’ll get a kick out of this
WID 1066828
kicker
[Using estimated or internet-retrieved values, groups use whiteboards while they are
guided through the problem. Have them check the group roles webpage for ponderables.]
Problem: You kick a soccer ball as it rolls by. Your kick is
perpendicular to its initial path. It ends up being deflected 20˚, but
stays on the ground. How hard did you kick it? First have them
sketch an overhead view. Place x to the left, z is toward the bottom.
Make sure it is correct.
Have students Google the mass of soccerball (between 350 and 450 g).
(http://www.fifa.com/aboutfifa/developing/pitchequipment/football/testcriteria.html)
and justify why they can believe their site. Go with 0.40 kg.
Have students estimate speed as it rolls by. This can be in comparison to how fast a
runner can go (~100 m in 10 s). Go with something like 3.5 m/s.
Have students estimate collision time. This is hard for them. Think about ball travel
during contact (not much). They should have already read the colliding students example
on pages 78-81.) Go with a couple of milliseconds (0.002 s).
Fshoe
System: Ball
Surroundings: earth, shoe
Momentum principle
ball
p f = pi + Fnet Dt
mvxf ,0, mvzf = mvxi ,0,0 + 0, Fgrass - FEarth , Fshoe Dt
vxf = vxi
mvzf
= Fshoe
Dt
Direction of v = v̂ = cos 20 ,0,cos110 = 0.9397,0,-0.342
v = 3.5,0,vzf
m
s
and
= v v̂ = v 0.9397,0,-0.342
(
) (
)
so v =
vx
v̂x
=
3.5 ms
0.9397
= 3.725 ms
vzf = 3.725 ms ´ -0.342 = -1.274 ms
Fshoe =
Chapter 2
mvzf
Dt
=
(
0.40 kg ´ -1.274 ms
0.002 s
) = -250 N
Fshoe = 0,0,-250 N
5
Check: Right units, reasonable value.
[Note: it's very important to take some time here to reinforce that there are two correct
solutions: one involving directional cosines and one using trig. They are related through
directional cosines, and many will find that approach simpler (and others, who have had
HS physics will want to use trig).]
tan 20 =
vzf
vzf
3.5
20˚
vxf = 3.5 m/s
What if we had kicked it upward at <2,3,0> m/s. What is its velocity in ½ s?
p f = pi + Fnet Dt
mvxf , mv yf ,0 = mvxi , mv yi ,0 + 0,-mg,0 Dt
vxf = vxi = 2 ms
(
)(
and v yf = v yi - gDt = 3 ms - 9.8 kgN 0.5 s
v = 2,-1.9,0
)
m
s
Tangible: Activity – Predict this!
WID 1074663
xye34
Situation 1: no fan, pushed (Fan Push.mov)
Situation 2: fan opposed start from rest (Fan on Forward.mov)
Situation 3: fan opposite to initial v (Fan on Backward.mov)
1) Watch the video for Fan Off and write a detailed description of what you just saw. Pay
attention to how the cart moved.
After group discussion, sketch predictions of what the following plots look like for this motion:
(Make sure your group has come to a consensus)
x vs. t
px vs. t
Fnet, x vs. t
Then do same thing for other two movies.
Students watch movies again and describe the motion for each situation. Then, predict
and draw (by hand) graphs of x vs. t, px vs. t, and Fnet, x vs. t and compare to other groups.
Don’t show them the following graphs until after the next activity, but you can have them
share their sketches.
Chapter 2
6
Fan Off
Fan Forward
Fan Backward
Chapter 2
7
note 2 slopes of Vx graph. Draw FBDs
Dr
vave. x = x
Dt
px = mvx
Fave. x =
-fan - friction
slope of x is vx
for low speeds
Dpx
Dt
slope of p x is Fave. x
friction - fan
Dpx mDvx
=
= max
Dt
Dt
-Ffan,x - Ffriction,x = m ( -0.25)
Fave. x =
mDvx ö
æ
-Ffan,x + Ffriction,x = m ç -0.3 =
8÷
è
Dt ø
subtract second from first equation, 0.54 kg cart:
-2Ffriction,x = 0.54 ( 0.13)
Ffriction,x = -0.035N
Chapter 2
Ffan, x = 0.17N
8
They will then compare these predictions to measurements and simulations in the
following activity (Tangible: Fan Cart 3-way Activity).
Tangible: Activity – Fancart 3-way Activity
Initially:
• If you are in an "a" group, do Lab: Fan Cart motion
• If you are in a "b" group, do Lab: Fan Cart movie analysis
• If you are in a "c" group, do VPython: Modeling Motion
Once you have completed your group's part, switch to another, then another.
Lab: Fan Cart motion
WID 1074677 doit
Previously, you made predictions about the graph of the x-component of position vs time
and momentum vs time for a cart on a track in cases of fan off, fan force in +x dir, and
fan force in –x direction (Activity – Predict this). Now, you are actually going to do the
experiment, using a motion detector that records the position of the moving object. The
data from the motion detector is fed into a handheld Vernier LabQuest device.
Note the devices need to be connected to the motion detector, and the AC power adaptor
needs to be plugged in too (didn’t have time to charge the batteries).
On tables, tracks with fan cart in center of table. Each group will record x vs t and v x vs t
for the three situations where you made predictions (Activity – Predict this!). Make plots
on the LabQuest screen. Each group makes these measurements. Be ready to help other
groups if they are having trouble using the equipment.
As you gather the plots, think and discuss with your group: Do these measurements
match your predictions?
Let’s discuss—how did your measured graphs differ from predictions? (Discuss in
particular transients at beginning and end while your hand touches the cart. Note for the
no fan case, the velocity isn’t constant—why? And what about the fan opposing initial v
case? Any difference in graph for first half vs. second half?)
Reason for difference in first vs. second half—draw the directions of the forces acting on
the cart for first half and second half. Come up with a way to calculate the frictional force
on the cart based on the data you’ve collected. (Hint: You’ll need the momentum
principle! Hint: You’ll need the mass of the fan cart!)
Chapter 2
9
You can convert velocity data to momentum data directly on LabQuest. Also, you can
measure the slope of the graph.
Lab: Fan Cart movie analysis
WID 1074679 doit
Fan Cart Videos - Data.xls
Video analysis.pdf
For the three situations where you made predictions (Activity – Predict this!), use Excel
data (or LoggerPro video analysis) to plot x vs. t and px vs. t and then compare to your
predictions. Instructions in WebAssign.
VPython: Modeling Motion
WID 1070724
VPython: Modeling Motion
fancart.py
You’ve analyzed this problem with theory (momentum
principle) and experiment. There is a third way to
tackle scientific problems—computer modeling.
You will create a VPython model of this system. This will introduce you to using
VPython to model motion (rather than just create static scenes as we did previously).
Open the WebAssign. Link to the instructions are there.
Checking Questions:
• Which statement represents the position update formula?
Note: Position update should always go after momentum principle
• Which statement represents the Momentum Principle?
• What would you have to change in your program to make the cart start at the right end
of the track and move to the left? Initial position and initial velocity
• Should the Momentum Principle statement be placed before the loop or inside the loop?
Why? Inside the loop
• What happens when the initial momentum of the fancart includes a +y component?
Clicker Questions up to springs, through question 2.3d
Tangible: Collisions
Note: this activity is designed to be fairly vague and open-ended.
Use one track, two carts (one with mass bars), and a scale per table. They will need to
Chapter 2
10
share data. Have the team that isn’t using LabQuest to get the cart and bar masses using a
triple-beam balance while other two teams set level the track, place the sensors at
opposite ends, and practice some runs. (Use two sonic rangers simultaneously to measure
speeds.) Compare total momentum afterwards to total momentum before. Guide them
through equal masses with one starting at rest. Can try two equal masses, heavy into light,
and light into heavy. Can even have them stick together if there is velcro on the carts.
Lab: Impulse
WID 624962 jolt
Lab using LabQuest and the force probe to relate impulse and change in momentum.
Checking Questions
• How does force of the cart on the spring compare to the force of the spring on the cart?
– The graph actually measures the force of the spring on cart, but they are
interested in the force of the cart on the spring. The two are equal and opposite.
• When Fx is the biggest it ever gets, what is px? Is px also at a maximum? Is px
proportional to Fx?
– Highlight the difference between px and ∆px
• How does the x component of the net impulse found using |F x dt| compare to the x
component of the change in the cart’s momentum?
– The connection between the two sides of the momentum principle
• What would you expect to happen with a stiffer spring, but the same change in
momentum?
– Area under the curve should be the same (∆p = Fx dt)
– Stiffer spring would compress less than the softer spring
– Time of impact would be less with the stiffer spring
– Peak force would be greater with the stiffer spring
– The Fx vs. t curve would be narrower and taller
Demo VPython: Baseball in flight
Baseball.py
Draw FBD at several places (hit, way up, top, way down). Look for “force of the hit.”
Chapter 2
11
Guide them through VPython, first with no air resistance, then with drag proportional to
speed squared. Problem 7.P.68 will show F~(8.8e-3)v2.
Discussion: Predicting motion with non-constant forces
We’ve seen how to use momentum principle to predict motion when net force is constant
(e.g. mg, fan): System, surroundings, initial and final time, update momentum, update
position.
Last time you saw an example of a program that predicted the motion of an object when
the force was NOT constant: air resistance on ball
Note that since the force is changing, it’s very useful to use a computer program to model
the system. Often it’s the only way to make these predictions!
We follow the same procedure, but we repeat it over multiple time steps (loop in a
computer program). Each time we are finding a new force, finding a new momentum,
finding a new position, repeating.
Tangible: Activity – Stretch your mind
WID 2296700 boing
Spring force law, pg 64
Fspring = ks s where s = L - L0 is the absolute value of the stretch
direction is to restore the spring to its relaxed length
Requires Pasco ME-8970, Equal Length Spring Set
A groups find a way to measure k for the Blue spring (30 N/m +/-10%)
B groups find a way to measure k for the White spring (40 N/m +/-10%)
C groups find a way to measure k for the Green spring (50 N/m +/-10%)
(fyi, Red spring is 25 N/m and Yellow spring is 35 N/m)
They should graph s vs F (not m) in Excel. (Remind them of scatter graph in Excel.) Ask
what the slope of stretch vs F is (and invert it). Get at force law this way.
Discussion: Springs
Talk through a qualitative analysis of spring-mass oscillation. Show them SuperSpring.py
Start a horizontal oscillator from rest with some stretch and go through a few time steps,
guide them through plotting x components of momentum and position vs. time (consider
equilibrium and extreme points). What do these curves look like?
Chapter 2
12
y
y
Initial condition
equilibrium
x
z
ri = xi ,0,0 m
x
z
pi = 0,0,0
Fnet = -ks xi ,0,0 Nxi
m
s
1
equilibrium
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
1
2
3
4
5
6
7
x
Fx
Px
equilibrium
0
-0.8
-1
Discuss analytical solution for an idealized system (no friction etc.)
Draw a cosine curve, define amplitude and period. Momentum principle says slope of p
vs t gives F graph.
Show why trig function is a valid solution for the ideal system, starting from the
momentum principle:
Chapter 2
13
dp
= Fnet
dt
dpx
= Fnet x = -ks x
dt
d
d2x
m
=m 2 =m
dt
dt
dvx
(
2
( Acos (w t )) = m d ( - Aw sin (w t )) = m - Aw cos (w t ) = -k x
(
)
dt
dt
s
2
( ))
( )
m - Aw 2 cos w t = -ks Acos w t
so mw 2 = ks and w =
ks
m
=
2p
T
x
Also useful to define f º
1
so w = 2p f
T
MAJOR POINT: In an elliptical orbit, the force has a component parallel to the
momentum that makes the momentum increase when approaching the star and decrease
when moving away. Since the force increases near the star, both the momentum and force
are biggest when nearest the star. But with the spring-mass oscillator, when is the
momentum biggest? Force biggest? Momentum smallest? Force smallest? Here the two
are anti-correlated. The momentum principle does NOT relate momentum directly to
force, rather it relates change in momentum to force.
Clicker Questions starting at springs
Chapter 2
14