Transcript Testing

Class #15 of 30
Exam Review

Taylor 7.50, 7.8, 7.20, 7.29,
ODE’s, 4.4, 4.7, 4.15, 4.19, 4.20
1 :72
Test #2 of 4
Thurs. 10/17/02 – in class


Bring an index card 3”x5”. Use both sides. Write
anything you want that will help.
You may bring your last index card as well.
Anything on last 3 homeworks
Lagrangian method (most of exam)
Line integrals / curls
Generalized forces / Lagrange multipliers /
constraint forces
Tuesday 10/15 – Real-time review / problem
session
2 :08
Taylor 7-50
A mass m1 rests on a frictionless horizontal table. Attached to it is a
string which runs horizontally to the edge of the table, where it
passes over a frictionless small pulley and down to where it
supports a mass m2. Use as coordinates x and y, the distances
of m1 and m2 from the pulley. These satisfy the constraint
equations f(x,y)=x+y=const. Write down the two modified
Lagrange equations and solve them for x’’, y’’ and the Lagrange
multiplier Lambda. Find the tension forces on the two masses.
x
f

y
m1
f

x
y
m2
f  x y
m1m2
  g
m1  m2
3
:12
Atwood’s Machine
Lagrange Multiplier Recipe -- CORRECTED


6) y1   g  a1; y2   g  a2
m1
m2
f

y1
 m1m2 
  2g 

m

m
 1 2
f

y2
 m1m2 
2g 

m1  m2 
 m2  m1 
m1
m2

6) y1 
g 
g
m1
 m1  m2 
f

 constraint force  T1  
y1
 m1m2 
  2g 
  mg ( for m1  m2 )
 m1  m2 
4 :40
y1
y2
Class #14 Windup - CORRECTED
L d  L 
f
    
qi dt  qi 
qi
f
FCi  
qi
L
 generalized force
q
L
 generalized momentum
q
Exam next Thursday
5 :72
ODE Summary
Math
For a, b  0
x  ax  x  x0e
Physics
v  3 D v  v  v0e 3 Dt
at
x  ax  x  x0e  at
x  bx  x  x1e
bt
x  bx  x  x1ei
 x2e 
bt
 x5 sin( bt   )
bt
 x2e i
 x3 sin( bt )  x4 cos( bt )
k
mx  kx  x  x5 sin( t   )
m
bt
g
l   g    6 sin( t   )
l
6 :55
Class #15 Windup
HW due
Thursday
Tues 3-5, Wed
4-5:30
Bring up to two
index cards
Midterm grades
will be posted
on web
Only one (hard)
HW problem
Happy 49ers!
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