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Physics 1D03
Harmonic Motion (
II
)
Serway 15.1—15.3
• Mass and Spring
• Energy in SHM
Practice: Chapter 15, problems 9,
15, 17, 19, 21, 67
Physics 1D03
Simple Harmonic Motion
SHM:
)
cos(
φ
ω
+
=
t
A
x
a(t) =
 ω
2
x(t)
dt
dv
a
dt
dx
v
=
=
,
differentiate:
and find that acceleration is proportional to displacement:
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Physics 1D03
Mass and Spring
x
m
k
dt
x
d
a

=
=
2
2
M
x
F = kx
ma
kx
F
=

=
Newton’s 2
nd
Law:
so
This is a
2
nd
order differential equation
for the function
x(t)
.
Recall
that for SHM,
a
= ω
2
x
, identical except for the proportionality
constant. So the motion of the mass will be SHM:
x(t) = A cos (
ω
t +
φ
),
and to make the equations
for acceleration match, we require that
m
k
=
2
,
or
m
k
=
(and
= 2
π
f
,
etc
.).
Physics 1D03
Mass and Spring
•
f
∝
(force parameter/inertia parameter)
½
•
The frequency is
independent of amplitude
.
•
The same results hold for a mass on a
vertical
spring (forces are spring plus gravity), as long
as the displacement is measured from the
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This note was uploaded on 12/14/2010 for the course ENGINEERIN 1D03 taught by Professor N.l.mckay during the Spring '10 term at McMaster University.
 Spring '10
 N.L.McKay

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