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 Dt
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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