15.1 motion of an object attached to a spring 15.1 hooke’s law 15.2
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15.1 Motion of an Object Attachedto a Spring
15.1 Hooke’s law
15.2 a
x=−
kmx
F
s=− kx
15.2 The Particle in Simple Harmonic Motion
15.3
15.4
15.5
15.6 Position versus time for an x(t) = A cos (t +
object in simple harmonic
motion
15.7
d 2x
dt 2=−
kmx
2 k
m =
d 2x
dt 2=− 2x
d 2x
dt 2=−A d
dtsin t +φ( ) =− 2A cos t +φ( )
15.2 The Particle in Simple Harmonic Motion, cont.
15.8
15.9
15.10
15.11
15.12
15.13 Period
22 ƒ
T
π π= =
T =
2π
mk
=2π
1ƒ
2T
π
= =
2T
π
=
=
km
dxdt
=−Addtcos t +φ( ) =−A sin t +φ( )
15.2 The Particle in Simple Harmonic Motion, cont.
15.14 Frequency
15.15 Velocity of an object in simple harmonic motion
15.16 Acceleration of an object in simple harmonic motion
15.17 Maximum magnitudes of velocity and acceleration insimple harmonic motion
15.18
ƒ =
1T
=12π
km
v =
dxdt
=−Asin( t+φ)
a =
d2xdt2
=− 2Acos( t + φ)
v
max=A=
km
A
a
max= 2A=
kmA
15.3 Energy of the SimpleHarmonic Oscillator
15.19 Kinetic energy of a simple K = ½ mv 2 = ½ m2 A2 sin2 (t + ) harmonic oscillator
15.20 Potential energy of a U = ½ kx 2 = ½ kA2 cos2 (t + )simple harmonic oscillator
15.21 Total energy of a simple E = ½ kA 2
harmonic oscillator
15.22 Velocity as a function of
position for a simple harmonic
oscillator v =±
km
A2 −x2( ) =± 2 A2 −x2
15.4 Comparing Simple Harmonic Motionwith Uniform Circular Motion
15.23 x(t) = A cos (t + )
15.5 The Pendulum
15.24
15.25
15.26 Period of a simple
pendulum
15.27
15.28 Period of a physical
pendulum
d 2θdt2
=−gLθ
g
L =
22
LT
g
π π
= =
22
2
d mgd
dt I
θ θ θ⎛ ⎞=− =−⎜ ⎟⎝ ⎠
22
IT
mgd
π π
= =
15.5 The Pendulum, cont.
15.29
15.30 Period of a torsional
pendulum
d 2θdt2
=−κIθ
2I
T πκ
=
15.6 Damped Oscillations
15.31
15.32
15.33
2
2
k b
m m ⎛ ⎞= −⎜ ⎟
⎝ ⎠
x =Ae−b 2m( )t cos (t+φ)
−kx−b
dxdt
=md2xdt2
15.7 Forced Oscillations
15.34
15.35
15.36 Amplitude of a driven
oscillator
( )
0
222 2
0
FmA
bmω
ω ω
=⎛ ⎞− + ⎜ ⎟⎝ ⎠
F∑ =ma → F0 sin t − b
dxdt
−kx=md2xdt2
x = A cos t+φ( )
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