Hooke’s Law - Linville Hooke's Law AP.pdf · • Hooke's Law gives the force acting on a spring...
Transcript of Hooke’s Law - Linville Hooke's Law AP.pdf · • Hooke's Law gives the force acting on a spring...
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Hooke’s
Law
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• Hooke's Law gives the force acting on a
spring or other elastic material when it has
been stretched or compressed
• The direction of the force is opposite to the
direction of stretch
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Weight of object stretches spring
Restoring force acts up
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force
k spring constant (N/m)
of spring
(change in length)
s
s
F k x
F restoring
x displacement
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Hooke’s Law (AP)
• Fs = restoring force
• x = displacement
sF k x
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• F is proportional to x
sF k x
Is the equation of a line
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Example
• A 400 g mass is hung form the lower end
of a spring. The spring stretches 0.200 m.
Calculate k.
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Solution
m/N.k
m.
kg/N.kg.k
x
gm
x
Fk
xkF
s
s
619
2000
8194000
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Example
• A 150 g mass is attached to one end of a
horizontal spring (k = 44.3 N/m) and the
spring is stretched 0.104 m.
a) Determine the maximum acceleration
when the mass is released.
x
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Solution
2
44.3 / 0.104
0.150
30.7 / left
s netF F
k x m a
k xa
m
N m ma
kg
a m s
Apply Newton’s
2nd Law
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b) Determine the acceleration of the object
when it is 0.055 m from the equilibrium
position
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2
44.3 / 0.055
0.150
16.2 / left
s netF F
k x m a
k xa
m
N m ma
kg
a m s
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• The magnitude of the acceleration is
constantly changing since the restoring
force changes as x changes
• Non-uniform accelerated motion means
you can’t use the kinematics equations
from unit 1
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Example
• A spring is connected to a box and its k
determined. The spring is then cut in half
and connected to the box as shown.
Describe the value of k for the two springs
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Solution
• It's 2k, since each spring now only extends
half the displacement of the original spring
when subjected to the same force.
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More than One Spring
• Two springs are
connected vertically in
series and a mass is
suspended from the
lower end. Derive an
expression for the
effective spring
constant.
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More than One Spring
• F1 = -k1x1
• F2 = -k2x2
• The forces on each
spring are equal. (ignore
the spring masses)
• -k1x1 = -k2x2
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More than One Spring
• -k1x1 = -k2x2
• F = -keff(x1 + x2)
• -keff(x1 + x2) = -k1x1 + -k2x2
2 21
1
k xx
k
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More than One Spring
𝑘𝑒𝑓𝑓𝑘2𝑥2𝑘1
+ 𝑥2 = 𝑘2𝑥2
Divide by x2
• 𝑘𝑒𝑓𝑓𝑘2
𝑘1+ 1 = 𝑘2
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More than One Spring
• 𝑘𝑒𝑓𝑓𝑘2
𝑘1+ 1 = 𝑘2
• 𝑘𝑒𝑓𝑓𝑘2+𝑘1
𝑘1= 𝑘2
• 𝑘𝑒𝑓𝑓 =𝑘2𝑘1
𝑘2+𝑘1
1 2
1 1 1
effk k k
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More than One Spring
• Two springs are
connected as shown and
a force is applied. What
is the spring constant of
the system? (connected
in parallel)
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More than One Spring
• The value of x is the
same for both springs
• F = -k1x + -k2x
• F = -(k1 + k2)x
• keff = k1 + k2
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Example • Two identical unstrained springs (k = 15
N/m) are connected to an 1.0 kg object as
shown. If the object is moved 4.0 cm to the
left and released, what is the acceleration
magnitude of the system?
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Spring Potential Energy
• the energy stored
in a spring
depends on the
displacement
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• the work done to stretch the spring = area
under graph
2
2
1
2
1
2
1
2
1
kxE
xkxE
FxE
abEwork
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AP
• Us = ½ kx2
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Practice
• P 301: 1, 2, 3, 4, 5