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# Spring Forces and Simple Harmonic Motion

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```Spring Forces and Simple
Harmonic Motion
Chapter 10
Expectations
At the end of this chapter, students will be able to:
пЃ¶ Apply HookeвЂ™s Law to the calculation of spring
forces.
пЃ¶ Conceptually understand how simple harmonic
motion is caused by forces and torques obeying
HookeвЂ™s Law.
пЃ¶ Calculate displacements, velocities, accelerations,
and frequencies for objects undergoing simple
harmonic motion.
Expectations
At the end of this chapter, students will be able to:
пЃ¶ Calculate the elastic potential energy resulting
from work done by spring forces.
пЃ¶ Understand how pendulums approximate simple
harmonic motion.
пЃ¶ Calculate the natural frequencies of both simple
and physical pendulums.
пЃ¶ Conceptually understand the ideas of damped,
driven, and resonant simple harmonic motion.
Expectations
At the end of this chapter, students will be able to:
пЃ¶ Analyze the elastic deformation of objects in
terms of stress, strain, and the elastic moduli of
materials.
пЃ¶ Express HookeвЂ™s Law in terms of stress and
strain.
Spring Forces
A spring resists
being stretched
or compressed.
Spring Forces
The force with
which the spring
resists is
proportional to
the distance
through which it
is compressed or
stretched:
F пЂЅ пЂ­ kx
вЂњspring constantвЂќ
SI units: N/m
Spring Forces вЂ“ HookeвЂ™s Law
F пЂЅ пЂ­ kx
This relationship is
called HookeвЂ™s
Law.
HookeвЂ™s Law
Robert Hooke
1635 вЂ“ 1703
English mathematician and
natural philosopher;
contemporary of Isaac
Newton
Early builder of microscopes
and telescopes
HookeвЂ™s Law
The direction of
the spring force
is always
opposite the
direction of the
stretching or
compression of
the spring вЂ“
hence, the minus
sign.
F пЂЅ пЂ­ kx
A Consequence of HookeвЂ™s Law
Consider NewtonвЂ™s second law and HookeвЂ™s Law
simultaneously:
F пЂЅ ma пЂЅ пЂ­ kx
a пЂЅ пЂ­
k
x
m
Now, a small and sneaky bit of calculus:
2
aпЂЅ
d x
dt
2
пЂЅпЂ­
k
x
пѓћ
x пЂЅ A cos пЃ· t
m
differential equation
its solution
Simple Harmonic Motion
Motion described by this equation:
x пЂЅ A cos пЃ· t
displacement (m)
time (s)
amplitude (m)
is called simple harmonic motion (SHM). It is:
пЃ¶ periodic (repeats itself in time)
пЃ¶ oscillatory (takes place over a limited spatial
range)
x пЂЅ A cos пЃ· t
Simple Harmonic Motion
displacement vs. пЃ· t
A = 1.00 m
1.00
0.80
0.60
displacement, m
0.40
0.20
0.00
-0.20
-0.40
-0.60
-0.80
-1.00
0.00
2.00
4.00
6.00
8.00
10.00
12.00
14.00
SHM: Reference Circle Representation
Y
A vector of magnitude
A rotates about the
origin with an angular
velocity пЃ·.
A cos пЃ· t
A
пЃ·t
X
The x component of
the vector represents
the displacement.
SHM: Frequency
Since there are 2p radians in each trip (вЂњcycleвЂќ)
around the reference circle, the вЂњcycleвЂќ frequency
is related to the angular frequency by
пЃ· пЂЅ 2p f
or
f пЂЅ
SI units of вЂњcycleвЂќ frequency, f:
cycles / s = Hertz (Hz)
пЃ·
2p
SHM: Velocity
Y
We can calculate the
velocity from the
reference circle
representation:
- vT sin пЃ· t
vT
пЃ·t
A
пЃ·t
X
vT пЂЅ r пЃ· пЂЅ A пЃ·
пЂ­ v T sin пЃ· t пЂЅ пЂ­ A пЃ· sin пЃ· t
v пЂЅ пЂ­ A пЃ· sin пЃ· t
SHM: Acceleration
Y
aC пЂЅ rпЃ· пЂЅ A пЃ·
2
2
a пЂЅ пЂ­ a C cos пЃ· t
-aC cos пЃ· t
пЃ·t
aC
пЃ·t
X
a пЂЅ пЂ­ A пЃ· cos пЃ· t
2
SHM: Velocity and Acceleration
пЃ¶
пЃ¶
пЃ¶
пЃ¶
пЃ¶
пЃ¶
пЃ· must be expressed in rad/s.
Like displacement, velocity and acceleration are
periodic in time.
Maximum velocity: v max пЂЅ A пЃ·
2
Maximum acceleration: a max пЂЅ A пЃ·
Acceleration has maximum magnitude at extremes
of displacement.
Velocity has maximum magnitude when
displacement is zero.
Mass on a Spring System
Natural frequency for a mass m on a spring with
spring constant k:
2
aпЂЅ
d x
dt
2
пЂЅпЂ­
k
x
x пЂЅ A cos пЃ· t
пѓћ
m
пЃ· пЂЅ 2p f пЂЅ
k
m
Work Done in Straining a Spring
Stretch or compress a spring by a displacement x
from its unstrained length.
Initial force: F0 = 0
Average force:
F пЂЅ
Final force: Fmax = kx
1
kx
2
Work over a displacement x: W пЂЅ F x пѓћ W пЂЅ
1
2
kx
2
Elastic Potential Energy
The spring force is a conservative force:
пЃ¶
пЃ¶
Like all conservative forces, its work is pathindependent.
Like all conservative forces, it is associated with a
form of stored or potential energy.
Elastic potential energy:
EPE пЂЅ
1
2
kx
2
Total Mechanical Energy
We add another term:
E пЂЅ KE пЂ« KE R пЂ« GPE пЂ« EPE
E пЂЅ
1
2
mv пЂ«
2
1
2
I пЃ· пЂ« mgh пЂ«
2
1
2
kx
2
The Simple Pendulum
A simple pendulum is a particle
attached to one end of a
massless cord of length L. It is
able to swing freely and
without friction from the other
end of the cord.
Its frequency:
пЃ· пЂЅ 2p f пЂЅ
g
L
L
The Physical Pendulum
A physical pendulum is any real
object (mass m) suspended a
distance L from its center of
gravity, able to swing freely
and without friction from the
suspension point.
Its frequency: пЃ· пЂЅ 2 p f пЂЅ
mgL
I
L
Physical-Simple Correspondence
Notice that a simple pendulum would have a
moment of inertia: I пЂЅ mL 2
Substitute: пЃ·
пЂЅ
mgL
I
пЂЅ
mgL
mL
2
пЂЅ
g
L
As a physical pendulum becomes a simple one, its
frequency вЂњcollapsesвЂќ to that of a simple
pendulum.
Small-Angle Approximation
The restoring torque on a
pendulum does not actually
have the HookeвЂ™s Law form:
пЃ±
L
пЃ±
T sin пЃ±
Result: the restoring torque
increases with angle, but at less
than a linear rate.
T cos пЃ±
пЃґ R пЂЅ TL sin пЃ± cos пЃ±
mg
Small-Angle Approximation
How much less?
пЃ±, В°
% under linear
0.10
0.0002%
1.0
0.02%
2.0
0.08%
5.0
0.51%
10
2.0%
Damped Oscillations
If the only force doing work on an object is the
spring force (conservative), its mechanical energy
is conserved. If frictional forces also do work, the
objectвЂ™s mechanical energy decreases, and the
SHM is called damped.
If the frictional force is just large enough to prevent
oscillation as the object reaches its equilibrium
position, it is called critically damped.
Driven Oscillations
If a driving force acts on an object in addition to a
HookeвЂ™s Law restoring force, the harmonic
motion of the object is called driven.
Example: a tree in a gusty wind.
Driven Oscillations: Resonance
If the driving force is periodic, and is applied at the
natural frequency of the oscillating object, the
work done on the object adds up over multiple
cycles of motion, and large-amplitude motion
results. This is called resonance. The natural
frequency is sometimes called the resonant
frequency.
Example: a person on a swing, being pushed by
another person.
Material Deformation: Everything is a
Spring
Solid materials are interconnected, microscopically,
by powerful intermolecular bonding forces.
These forces behave like springs вЂ¦ with really large
spring constants.
Because of them, material objects resist
deformations, such as compression, elongation, or
shearing.
Tension and Compression
A force acts to increase the length of an object:
fractional change in
length
applied force
F пЂЅY
пЃ„L
L0
A
cross-
YoungвЂ™s modulus
sectional
SI units = N/m2
area
Thomas Young
1773 - 1829
English physicist,
physician, and
Egyptologist
Famous mostly for his
work in optics
Shear
A pair of forces act to shear an object (deform it slantwise):
applied force
F пЂЅ S
Shear modulus
SI units = N/m2
пЃ„X
L0
A
cross-sectional
area
Shear
1945 - ?
Has only one name
English physicist and pop
musician
Inventor of the shear
modulus
Rumored to have appeared in the 1983 version of Dune
Volume Deformation
In order to discuss volume deformation, it is necessary
to define a new force-related quantity: pressure.
Pressure is the ratio of the magnitude of a force
applied perpendicular to a surface to the area of that
surface:
P пЂЅ
F
A
SI units: N/m2 = Pascals (Pa)
Blaise Pascal
1623 вЂ“ 1662
French mathematician
Invented the first digital
calculator (the вЂњPascalineвЂќ)
Volume Deformation
A change in pressure changes
the volume of an object:
pressure
change
пЃ„P пЂЅ пЂ­ B
пЃ„V
V0
fractional
bulk modulus
change in
SI units: N/m2
volume
Stress and Strain
Stress is the deforming force applied to an object,
divided by its cross-sectional area:
stress пЂЅ
F
A
Stress has SI units of N/m2 (just as pressure and the
elastic moduli have).
Stress and Strain
Strain is the change in a dimensional quantity
expressed as a fraction of its un-deformed value:
strain пЂЅ
пЃ„L
L0
or strain пЂЅ
пЃ„X
or strain пЂЅ
L0
пЃ„V
V0
Strain is a dimensionless, unitless ratio.
Strain is the result of stress on a material object.
Stress and Strain
Consider the defining equation for YoungвЂ™s modulus:
F пЂЅY
пЃ„L
L0
A
Rearrange:
F
A
пЂЅY
пЃ„L
L0
To restate: stress пЂЅ пЂЁelastic modulus пЂ©пЂЁstrain пЂ©
This is the stress-strain formulation of HookeвЂ™s Law.
Summary: Elastic Moduli
deformation
length
elastic modulus
YoungвЂ™s modulus (Y)
shear
shear modulus (S)
volume
bulk modulus (B)
all moduli have the same SI units: N/m2 = Pa
equation
F пЂЅY
пЃ„L
A
L0
F пЂЅ S
пЃ„X
L0
пЃ„P пЂЅ пЂ­ B
пЃ„V
V0
A
```
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