Chapter 13 Simple Harmonic Motion

1
Chapter 13 Simple Harmonic Motion In
chapter 13 we will study a special type of motion
known as simple harmonic motion (SHM). It is
defined as the motion in which the position
coordinate x of an object that moves along the
x-axis varies with time t as x(t)
Asin(?t) or x(t) Acos(?t)
The condition for having simple harmonic
motion is that the force F acting on the moving
object is F -kx
(13-1)
2
Periodic motion is defined as a motion that
repeats itself in time. The period T ? the time
required to complete one repetition Units s A
special case of periodic motion is known as
harmonic motion. In this type of motion the
position x of an object moving on the x-axis is
given by x(t) Asin(?t) or
x(t) Acos(?t) The most general expression for
x(t) is
The parameters A, ?, and ? are
constants
x(t) Asin(?t ?)
(13-2)
3
Plots of the sine and cosine functions. Make
sure that you can draw these two functions
?/2 90º ? 180º 3?/2 270º 2? 360º Note
The period of the sine and cosine function is
2? (13-3)
4
Most general form of simple harmonic motion
x(t) Asin(?t ?)
A ? amplitude ? ? angular frequency ? ? phase
sin(a b) (sina)(cosb) (cosa)(sinb)
? x (Acos ?)sin?t (Asin?)cos?t The
coordinate x is a superposition of sin?t and
cos?t functions Special case 1 ? 0
? x A sin?t Special case 1 ?
?/2 ? x A cos?t (13-4)
5
The effect of the phase ? on is to shift the
Asin(?t ?) function with respect to the
Asin(?t) plot This is shown in the picture below

Red curve Asin(?t)
Blue curve Asin(?t ?)
(13-5)
6
Determination of A and ? These two
parameters are determined from the initial
conditions x(t 0) and v(t 0)
(13-6)
7
x Asin(?t)
v A?cos(?t)
Plots of x(t), v(t) and a(t) versus t for ?
0
a -?2Asin(?t)
(13-7)
8
Consider a point P moving around O on a circular
path of radius R win constant angular velocity ?.
The Cartesian coordinates of P are x and y.
The angle ? ?t. From triangle OAP we have OA
x Rcos?t and AP Rsin?t
The projection of point P on either the x- or the
y-axis executes harmonic motion as point P is
moving with uniform speed on a circular orbit
(13-8)
P
A
?
9
A mass m attached to a spring (spring constant
k) moves on the floor along the x-axis without
friction
(13-9)
10
(13-10)
11
(13-11)
12
Summary The spring-mass system obeys
the following equation

The most general solution of the equation of
motion is Note 1 The
angular frequency ? ( and thus the frequency f ,
and the period T) depend on m and k only

Note 2 The amplitude A and the phase ? on
the other hand are determined from the initial
conditions x(t 0) and v(t0)
(13-12)
x Asin(?t ?)
13
Energy of the simple harmonic oscillator
(13-13)
14
X
(13-14)
15
Why all this fuss about the simple harmonic
oscillator? Consider an object of energy E
(purple horizontal line) moving in a potential U
plotted in the figure using the blue solid line
E
Motion in the x-axis is allowed in the regions
where These regions
are color coded green on the x-axis. Motion is
forbidden elsewhere (color coded red) If we have
only small departures from the equilibrium
positions x01 and x02 we can approximate U by
the potential of a simple harmonic oscillator
(the blue dashed line in the figure)

(13-15)
16
We expand the function U(x) around one of the
minima (x01 in this example) using a Taylor series
E
If we choose the origin to be at xo1 ? U(x)
kx2/2 This approximation for U is indeed the
potential of a simple harmonic oscillator (dotted
blue line) (13-16)


17
Simple pendulum A mass m suspended from a
string of length ? that moves under the influence
of gravity
s
(13-17)
18
Simple Pendulum This differential
equation is too difficult to solve. For this
reason we consider a simpler version making the
small angle approximation
(13-18)
19
(13-19)
20
Summary The simple
pendulum (Small angle approximation)
? ltlt 1
(13-20)
21
In the small angle approximation we assumed that
? ltlt 1 and used the approximation sin? ? ?
We are now going to decide what is a small
angle i.e. up to what angle ? is the
approximation reasonably accurate? ?
(degrees) ?(radians) sin? 5 0.087 0.087 10
0.174 0.174 15 0.262 0.259 (1
off) 20 0.349 0.342 (2 off) Conclusion If
we keep ? lt 10 we make less that 1
error (13-21)
22
O
Energy of a simple pendulum
C
A
s ??
U 0
B
(13-22)
23
Energy of a simple pendulum
(13-23)
24
O
The Physical Pendulum is an object of finite size
suspended from a point other than the center of
mass (CM) and is allowed to oscillate under the
influence of gravity. The torque about point O
due to the gravitational force is
r
?
C
A
d
mg
(13-24)
25
O
r
?
mg
(13-25)
26
O
(13-26)
r
?
mg
27
Overall Summary
k
?
r
(13-27)
28
Damped Harmonic Oscillator Consider the
spring-mass system in which in addition to the
spring force Fs -kx we have a damping force
Fd -bv. The constant b is known as
the damping coefficient In this example Fd
is provided by the liquid in the lower part of
the picture
-kx
v
x
-bv
(13-28)
29
(13-29)
A plot of the solution is given in the figure.
The damping force . Results in an exponential
decrease of the amplitude with time. The
mechanical energy in this case is not conserved.
Mechanical energy is converted into heat and
escapes.
30
Driven harmonic oscillator
k
Consider the spring-mass system that oscillates
with angular frequency ?o (k/m)1/2
If we drive
with a force F(t) Fosin?t (where in
general ? ? ?o) the spring-mass system will
oscillate at the driving angular frequency ?
with an amplitude A(?) which depends strongly on
the driving frequency as shown in the figure to
the right. The amplitude has a pronounced
maximum when ? ?o This phenomenon is called
resonance

(13-30)
31
The effects of resonance can be quite dramatic as
these pictures of the Tacoma Narrows bridge taken
in 1940 show. Strong winds drove the bridge in
oscillatory motion whose amplitude was so large
that it destroyed the structure. The bridge was
rebuilt with adequate damping that prevents such
a catastrophic failure
(13-31)