Why study springs?
Springs are_____________. This means that if you:
a/ apply a ___________to them, and
b/ they become ____________ (bent out of shape), and
c/ then you remove the ___________, then they
d/ __________________ .
Many things have some elasticity, and so they
behave like springs:
•
wood
_____________
plastic
•
concrete
humans
water
•
air
the sun
atoms
•
quartz
speakers
water
•
______________________________
strings, air, drums…
Many elastic objects obey…
elastic
deformed
force
forced
bounce back
metal
musical instruments
___________ Law:
The compression or elongation
x
of an
ideal spring from its ________________ position (x = 0)
is ____________________________to the applied force
F
s
.
F
s
=
stretching or elongation:
compression:
More F
more ____________ or __________________.
x = 0
x = 0
F
s
=
F
s
x
F
s
x
directly proportional
kx
stretch
compression
Hooke's
equilibrium
Hooke's Law is often written:
F
s
=

kx
This is because it also describes the force that the
_______________ exerts on an ___________ that is attached
to it. The negative sign indicates that the direction of
the spring force is always _____________ to the
displacement of the object
spring itself
stretched
spring:
compressed
spring:
object
opposite
undisturbed
spring
______________
position, F
s
= __
equilibrium

x
+x
F
s
F
s
x = 0
0
F
s
___ 0
F
s
___ 0
>
<
Ex. A weight of 8.7 N is attached to a spring that
has a spring constant of 190 N/m. How much
will the spring stretch?
Equation
:
Given:
Unknown:
x
w/o weight
8.7
N
w/ weight
190 N/m
8.7 N
x = ?
k =
F
s
=
F
s
= kx
8.7 N = (190 N/m) x
x = 4.6 x 10

2
m
F
s
= kx
F
s
x
What quantity does the slope represent?
slope =
=
Compare to F
s
= kx
Solve for F
s
/x =
The slope represents _______________________________
direct
Ex: A force of 5.0 N
causes the spring to
stretch 0.015 m.
How far will it stretch
if the force is 10 N?
2 (0.015 m)
= 0.030 m
D
y/
D
x
F
s
/x
the spring constant,
k
.
k
5
10
.015
?
What are the units of the spring constant, k?
Solve…
F
s
= kx
…for k: k =
units of k: [k] = [ ]/[ ]
= (derived)
This can be also seen from the graph:
F
s
(N)
x (m)
k = the slope =
D
y/
D
x
So k
has the same
units as
D
y/
D
x:
F
s
/x
F
s
x
N/m
N/m
F
s
x
stiffer spring
_________ slope
_________ k
spring
A
spring
B
Ex. Comparing
two springs that
stretch different
amounts.
x
B
Applying the same
force F to both springs
Which spring stretches more?
Which is stiffer?
x
A
A
B
greater
larger
____________ PE

the energy stored in a spring when work
is done on it to stretch or compress it
PE
s
=
Ex. A spring with a spring constant of 370 N/m is
stretched a distance 6.4 x 10

2
m. How much elastic
PE will be stored in the spring?
How much work was done to stretch the spring by
this amount?
(½)kx
2
Elastic
____________ PE

the energy stored in a spring when work
is done on it to stretch or compress it
PE
s
=
Ex. A spring with a spring constant of 370 N/m is
stretched a distance 6.4 x 10

2
m. How much elastic
PE will be stored in the spring?
PE
s
= (½)kx
2
= (0.5)( 370 N/m)(6.4 x 10

2
m)
2
= 0.76 (N/m)(m
2
)
= 0.76 Nm
= 0.76 J
How much work was done to stretch the spring by
this amount?
W =
D
PE = 0.76 J
(½)kx
2
Elastic
PE
s
=
What happens to PE
s
when you double x?
The PE
s
quadruples.
When you triple x?
9x more PE
s
.
Ex: The elastic PE stored in a spring is 0.70 J when
it is stretched 0.010 cm. If the same spring is stretched
0.030 cm, how much PE will then be stored in it?
PE
S
x
prop. to
square
x changes from 0.010 to 0.030
it triples
9x more PE
s
9x (0.70 J) = 6.3 J
(½)
k
x
2
Ex: Plot F vs. x for
an ideal spring
F
x
What does the grey area represent?
area = (½)bh =
=
=
=
It represents the ____________ on the spring, and
the ______________________ in it.
(½)x
F
(½)x(
kx
)
(½)kx
2
PE
s
work done
W =
D
PE
s
energy stored
Ex: By looking at the area, you can see why the PE is
proportional to the ___________ of the displacement x:
F
1 x
x
2 x
3 x
1 x
2 x
3 x
____ triangle
area = ____
so PE = ____
____ triangles
area = ____
so PE = ____
____ triangles
area = ____
so PE = ____
A
1
A
A
square
4
4A
4A
9
9A
9A
1
2
3
4
One last warning:
If a spring is stretched too much, it ____________
permanently, and Hooke's Law (F
s
= kx ) _________
______________
Ex: Which of the graphs below shows a spring that
obeys Hooke's Law?
F
x
F
x
F
x
F
x
YES
and
NO
deforms
is no
longer valid.
A/
B/
C/
D/
Open your
Review Book packet
to pages: 82

3
Do problems #
39

56
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