Transcript Using physcics when the force is complicated…

Using physics when the force is
complicated…
If you’re lucky, you can solve Fnet = ma for the new r(t)
But, in general F(t) is also a function of time, and you can’t
find an equation for r(t).
Examples:
•modeling molecular dynamics
•Rocket propulsion
•Bacterial motion (Listeria propulsion by comet tails)
•Almost any real world problem!
What then?
First, let’s see how you can solve a
harder case of time varying force
• Wind resistance and drag forces do have
analytical solutions for position and velocity as
a function of time!
• We can also use computers to model them in
general
• If we use computers, then we can have
arbitrarily complicated forces!
• Let’s see how all this works for drag, rocketry
and a real research-grade problem…
Rocket Propulsion
•First, let’s talk through
what forces the space
shuttle experiences as a
function of time:
•List all the forces the
shuttle experiences
•Write down qualitatively
how these vary as a
function of time
•Then, we’ll try to
implement then for real…
Let’s see how a computer program can
do just this, for ANY force…
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Aspects of programming
Vpython
Parameters
Control structures (loops, flow control, logic)
What “=“ means in a program
Our basic approach
• We know how to solve the problem when Fnet
= constant.
• Then x = xo + voDt + ½ a Dt2
• And so on for y and z.
• How about finding x(t) for a very small time Dt
• Then, during Dt, force is still (pretty much)
constant
• Then, compute the new force and repeat!
How to read the program:
First part just sets things up and
describes what I am doing:
# 3D animation of rocket taking off from a flat Earth
# Version 1.0 9-25-09
# Plots the trajectory of the rocket and then its distance and speed vs. time
# No wind resistance in Version 1.0
from visual import *
from visual.graph import * # import graphing features
Any line that starts with # is just a comment
The computer does nothing when it sees these—they are
just to remind us what is going on!
The last bits grab useful functions from a master program
# initialization section
# Set up the values I will need
Next, list what we
know
Fthrust = 32.e6 # thrust force in Newtons (average)
dt = 1
# timestep in seconds
m = 2.0e6
# mass in kilograms
Setup Thrust,
Ntimesteps = 100
mass, time step size
# total number of timesteps for the simulation
Time = (# timesteps) X dt
g = 9.8
# acceleration of gravity--constant at first for a flat Earth
v_yo = 0.
v_xo = 0.
v_zo = 0.
# initial speed along y axis--zero at launch
# later maybe we will make it 2D!
# or even 3D!
Initial values
x_o = 0
y_o = 0
z_o = 0
# initial value of x, y and z--starts at the origin
of (vo, vo , vo ) at t = 0
Initial values of (xo, yo , zo ) at t = 0
This part sets up our plotting and
graphics
rocket = cone (pos=(x_o,y_o,z_o), radius=1000, axis=(0,5000,0), color=color.white)
# initialize rocket's position at (xo,yo,zo), pointing along +y
rocket.velocity = vector(v_xo,v_yo,v_zo)
# initialize rocket's velocity too
rocket_position=[]
rocket_speed=[]
The shuttle is plotted as a white cone
Its position vector is rocket.pos
It’s velocity vector is rocket.velocity
c = curve(pos=(x_o,y_o,z_o),color=color.red)
# plots the rocket's trajectory
A red curve indicates the shuttle’s trajectory
Now, compute the new position for
each time step (while force is
approximately constant)
n=0
Do what’s in the box while n < Ntimesteps
while n <= Ntimesteps:
rate (20)
F_net_y = Fthrust - m*g
# screen refresh rate
n is called an index
It counts off time steps
t = n dt is the time
First find Fnet
rocket_force = vector(0.,F_net_y,0.)
# at first we ignore the loss of mass from fuel ejection
rocket.velocity = rocket.velocity + (rocket_force/m)*dt
rocket.pos = rocket.pos + rocket.velocity*dt
# first update velocity, then update position second!
rocket_position.append(mag(rocket.pos))
rocket_speed.append(mag(rocket.velocity))
c.append(pos=rocket.pos)
n=n+1
Then use it to update
velocity and position
Here I just keep track of all the values I’ve computed to plot later
Finally, we plot the distance and speed
vs. time
# a canned functional plotting program from the vpython manual--use to plot distance and
speed vs. time
scene2 = display(title='Graph of position and speed')
#scene2=display() # new screen with the plots
funct1 = gcurve(color=color.cyan)
funct2 = gcurve(delta=0.05, color=color.yellow)
n=0
It plots shuttle position in blue
Shuttle velocity in yellow
Don’t worry about how this
works in detail!
while n <= Ntimesteps:
funct1.plot(pos=(n*dt,rocket_position[n])) # curve
funct2.plot(pos=(n*dt,40.*rocket_speed[n])) # vbars
n=n+1
OK let’s run the program!
How close is our final value of rocket
speed to NASA’s measured value?
Still not right…off by quite a bit.
What else did we leave out?
Hmmm…
Can you guess?
Take a minute or two to discuss and
propose some ideas!
OK, include wind resistance (Version 2)
New Setup information:
air_density = 1.0
altitude!
# air density kg/m^3 < --- correct for air pressure falling off with
drag_coefficient = 0.5 # values of 0.5 to 0.75 quoted in misc. rocketry sources as the
appropriate drag for this geometry (0.5 for shuttle)
area = 400
# in m^2
And now the force is different:
F_drag = -0.5*air_density*drag_coefficient*area*mag(rocket.velocity)*mag(rocket.velocity)
F_net_y = F_thrust - m*g + F_drag
OK, now account for loss of mass as
the fuel burns (Version 3.0)
Yet more setup:
SMME = shuttle main engine (uses liquid fuel)
Solid Boosters use a different fuel source
Use Dm = Dt dm/dt
m = m - (dm_dt_solid+dm_dt_SMME)*dt
# reduce the rocket mass by the ejected fuel
Effect of changing mass?
Newton’s law is really about the law of conservation
of momentum
If there is no force (if a system is isolated) then:
dptotal/dt = 0
Consider two objects, M and m (the rocket and its exhaust).
Then:
ptotal = procket + pexhaust
From this we see:
dptotal/dt = dprocket/dt + dpexhaust/dt = 0
Or:
Or:
dprocket/dt = - dpexhaust/dt
Dprocket = - Dpexhaust
How to include the effect of changing
mass?
Now consider what happens when the rocket expels
exhaust to move forward
V
Before: ptotal = (M + m) V
After:
m
ptotal = M (V + v) + m(V – vexhaust)
V - vexhaust
m
And before
: Dprocket = - Dpexhaust
M v = mvexhaust
(here vexhaust is a positive speed)
M
V+v
M
Main lessons:
1) Forces inside the system are equal and opposite:
The rocket exerts the same force on exhaust as the exhaust exerts
on the rocket (and both move in response)
2) The Thrust (force on the rocket) is determined by the exhaust
speed and the exhaust expelled per unit time:
Fthrust = dpexhaust/dt = d(mvexhaust)/dt = vexhaustdm/dt
3) We need to know something more to solve for both the final
rocket and exhaust speeds—you have to be given vexhaustdm/dt
4) Of course, the rocket’s mass changes as a function of time, and
the thrust force can also have many different dependences on
time
Finally, the thrust actually varies with
time… Version 4.0
F_thrust = F_thrust_solid*(1-(F_thrust_solid-14.e6)*(n/Ntimesteps)) + (F_thrust_SMME*
m_SMME/liquid_mass)
# thrust varies due to the actual solid booster thrust vs. time variation: 32MN to 14 MN linear
decrease over 110s, then sharp falloff to zero to end.
Still not right…off by quite a bit.
What else did we leave out?
Hmmm…
Can you guess?
Take a minute or two to discuss and
propose some ideas!
Gravity changes with altitude & Air density
decreases with altitude!
Versions 5.0 & 6.0
How big an effect are each of these?
Let’s see how well it works now!
air_density = exp(-mag(rocket.pos)*0.00011856)
Using measured data!
# correction for altitude
Listeria motion
• Listeria monocytogenes is a pathogen
(disease-causing) bacterium
• It moves by producing a “comet tail” of
filamentous actin molecules
• Experimentalists have collected movies of
listeria trapped in a 2D sandwich between
microscope slides and coverslips.
• They move in complex ways—can we model
their motion?
Actual Listeria trails (Shenoy et al. PNAS
2007)—movies @ Julie Theriot’s website
• White scale bars are 5
microns long
• At full speed it travels
at 0.2 microns/s
• Samples of its complex
trails
• Let’s play
Newton/Kepler!
• Can we find a simple
model that produces
all these curves from a
few parameters?
Actual Listeria trails (Shenoy et al.
PNAS 2007)—or see Julie Theriot’s
website
• Samples of its
complex trails:
• White scale bars
are 5 microns
long
• At full speed it
travels at 0.2
microns/s
There are competing ideas about its
motion…can we really model such a
complex system?
Theriot lab
How to model physically?
• Theriot’s group has proved
that Listeria rotates as it
moves along.
• If a bead is attached to the
bacterium’s side, the bead
rotates around as the
bacterium propels itself.
• Let’s model its propulsion
mechanism like it has a
little rocket attached at one
end set at a fixed angle!
• This rocket rotates about
the listeria’s body axis as it
moves along
How would you do this?
• Try to describe (maybe graphically, maybe
mathematically) how you would change our
program to model Listeria
• Then, let’s try out a real program!