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April 14, 2006
Physics for Scientists&Engineers 1
1
Physics for Scientists &
Physics for Scientists &
Engineers 1
Engineers 1
Spring Semester 2006
Lecture 47
April 14, 2006
Physics for Scientists&Engineers 1
2
Review: SHM
Review: SHM
!
Hooke’s Force Law:
!
Equation of motion:
!
Solution:
or:
!
Amplitude:
Phase:
!
Angular Frequency:
F
=
!
kx
d
2
x
dt
2
+
k
m
x
=
0
x
(
t
)
=
A
sin(
!
0
t
)
+
B
cos(
0
t
)
x
(
t
)
=
C
sin(
0
t
+
"
0
)
C
=
A
2
+
B
2
0
=
arctan(
B
/
A
)
0
=
k
m
m
d
2
x
dt
2
=
!
kx
April 14, 2006
Physics for Scientists&Engineers 1
3
Review: SHM (2)
!
Period:
!
Frequency:
T
=
2
0
=
2
k
/
m
=
2
m
k
f
=
0
2
=
1
2
k
m
April 14, 2006
Physics for Scientists&Engineers 1
4
Work and Energy in Harmonic Oscillations
!
We have calculated the functions v(t) and x(t) that describe
the oscillation of a mass on a spring in time
!
We can use these to find the potential energy stored in a
spring as a function of time
!
And the kinetic energy as a function of time
where we have used
U
=
1
2
kx
2
=
1
2
kC
2
sin
2
(
0
t
+
#
0
)
K
=
1
2
mv
2
=
1
2
m
0
2
C
2
cos
2
(
0
t
+
0
)
=
1
2
kC
2
cos
2
(
0
t
+
0
)
0
2
=
k
m
x
(
t
)
=
C
sin(
0
t
+
0
)
v
(
t
)
=
0
C
cos(
0
t
+
0
)
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Physics for Scientists&Engineers 1
5
Work and Energy in Harmonic Oscillations (4)
!
Combining these we find the total Energy:
using
!
The total Energy is
constant, as found
previously.
!
It oscillates between
K(t) and U(t).
E
=
K
+
U
=
1
2
kC
2
cos
2
(
"
0
t
+
#
0
)
+
1
2
kC
2
sin
2
(
0
t
+
0
)
=
1
2
kC
2
cos
2
+
sin
2
=
1
April 14, 2006
Physics for Scientists&Engineers 1
6
Quiz 46.1
!
The driver of a car decides to smoke a cigar.
!
Which window should be opened so that the
passenger, sitting beside the driver, does not have
to breathe the secondhand smoke?
A) The driverside window
B) The passengerside window
April 14, 2006
Physics for Scientists&Engineers 1
7
Pendulum Motion
Pendulum Motion
!
We have asserted that the angle
!
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 Physics, Force

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