Transcript Simple harmonic motion

Chapter 12
Simple Harmonic Motion
Photo by Mark Tippens
A TRAMPOLINE exerts a restoring force on the
jumper that is directly proportional to the average
force required to displace the mat. Such restoring
forces provide the driving forces necessary for
objects that oscillate with simple harmonic motion.
After finishing this section,
you should be able to:
• Write and apply Hooke’s Law for objects moving
with simple harmonic motion.
• Write and apply formulas for
finding the frequency f, period T,
velocity v, or acceleration a in
terms of displacement x or time t.
• Describe the motion of pendulums
and calculate the length required
to produce a given frequency.
Periodic Motion
Simple periodic motion is that motion in which a
body moves back and forth over a fixed path,
returning to each position and velocity after a
definite interval of time.
1
f 
T
Amplitude
A
Period, T, is the time
for one complete
oscillation. (seconds,s)
Frequency, f, is the
number of complete
oscillations per
second. Hertz (s-1)
Example 1: The suspended mass makes 30
complete oscillations in 15 s. What is the
period and frequency of the motion?
15 s
T
 0.50 s
30 cylces
x
F
Period: T = 0.500 s
1
1
f  
T 0.500 s
Frequency: f = 2.00 Hz
Simple Harmonic Motion,
SHM
Simple harmonic motion is periodic motion in
the absence of friction and produced by a
restoring force that is directly proportional to
the displacement and oppositely directed.
x
F
A restoring force, F, acts in
the direction opposite the
displacement of the
oscillating body.
F = -kx
Hooke’s Law
When a spring is stretched, there is a restoring
force that is proportional to the displacement.
F = -kx
x
m
F
The spring constant k is a
property of the spring given by:
k=
DF
Dx
Hooke's law is the relationship
between the force exerted on the
mass and its position x
Example 2: A 4-kg mass suspended from a
spring produces a displacement of 20 cm.
What is the spring constant?
The stretching force is the weight
(W = mg) of the 4-kg mass: 20 cm
F = (4 kg)(9.8 m/s2) = 39.2 N
F
m
Now, from Hooke’s law, the force
constant k of the spring is:
k=
DF
Dx
=
39.2 N
0.2 m
k = 196 N/m
Period and Frequency as a
Function of Mass and Spring
Constant.
For a vibrating body with an elastic restoring force:
Recall that F = ma = -kx:
1
f 
2
k
m
m
T  2
k
The frequency f and the period T can be found if
the spring constant k and mass m of the vibrating
body are known. Use consistent SI units.
Example 3: The frictionless system shown
below has a 2-kg mass attached to a
spring (k = 400 N/m). The mass is
displaced a distance of 20 cm to the right
and released.
What is the frequency of athe motion?
v
x
m
x = -0.2 m
1
f 
2
x=0
k
1

m 2
f = 2.25 Hz
x = +0.2 m
400 N/m
2 kg
Summary
Simple harmonic motion (SHM) is that motion in
which a body moves back and forth over a fixed
path, returning to each position and velocity
after a definite interval of time.
The frequency (rev/s) is the
reciprocal of the period (time
for one revolution).
x
m
F
1
f 
T
Summary (Cont.)
Hooke’s Law: In a spring, there is a restoring
force that is proportional to the displacement.
F  kx
x
The spring constant k is defined by:
m
F
DF
k
Dx
Summary: Period and
Frequency for Vibrating
Spring.
x
a
v
m
x = -A
1
f 
2
x=0
k
m
x = +A
m
T  2
k
Summary: Simple Pendulum
1
f 
2
g
L
L
L
T  2
g