elastic potential energy. equations the elastic potential energy of a spring is one half the product...
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Elastic potential energy
Elastic potential energy
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EquationsEquations
The elastic potential energy of a spring is one half the product of its spring constant multiplied by the square of its extension or compression.
or
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SpringsSprings
free length
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Force and deformationForce and deformation
When you apply a force to a spring, it deforms.
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WorkWork
The applied force does work on the spring.
The change in the spring’s length is called the deformation, x.
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Elastic potential energyElastic potential energy
The work done to stretch or compress the spring is stored in the spring as elastic potential energy.
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EquationsEquations
The elastic potential energy of a spring is one half the product of its spring constant multiplied by the square of its deformation.
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What is the spring constant k ?What is the spring constant k ?
The spring constant tells you the stiffness of the spring.
The spring constant k is a property of the spring itself. It does not change when the spring is deformed.
• Stiff springs have high spring constants.
• Weak springs have low spring constants.
k
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Units of the spring constantUnits of the spring constant
The spring constant has units of N/m, or newtons per meter.
• Example: A 300 N/m spring requires 300 N of force to stretch 1 meter.
• A stiff spring needs a large force to stretch it a meter, so it has a large spring constant.
• A stiff spring stores more potential energy per meter of stretch.
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• It can be positive or negative.
• It points in the opposite direction of the spring force.
• It has units of meters.
The deformation x is the change in the length of the spring.
What is x ?What is x ?
x
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Perfect for a mountain bike!Perfect for a mountain bike!Inside the fork tube is a spring with a spring constant of roughly 20,000 N/m.
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Calculating forceCalculating force
How much force is needed to compress this spring one centimeter?
k = 20,000 N/m
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Hooke’s lawHooke’s law
The spring pushes back in the opposite direction with a force of -200 N.
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Elastic potential energyElastic potential energy
Where does this formula come from?
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Elastic potential energyElastic potential energy
Hypothesis: The elastic potential energy is derived from the work done to deform the spring from its free length . . .
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WorkWork
Work is force times distance.
W = Fd
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W = Fd
F = -kx
Hooke’s lawHooke’s law
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W = Fd
Hooke’s lawHooke’s law
F = -kxwhere k is the spring constant in N/m . . .
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F = -kx
W = Fd
Hooke’s lawHooke’s law
where k is the spring constant in N/m . . .
and x is the change in length of the spring in meters.
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Force vs. distanceForce vs. distance
BUT the force F from a spring is not constant.
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On a graph of force vs. distanceit is a line of constant slope.
Force vs. distanceForce vs. distance
BUT the force F from a spring is not constant.
It starts at zero and increases as the deformation x increases.
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Force vs. distanceForce vs. distance
The area on this graph . . .
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Force vs. distanceForce vs. distance
The area on this graph is force times distance . . .
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The area on this graph is force times distance which is the work done!
Force vs. distanceForce vs. distance
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Deriving the equationDeriving the equation
The area of this triangle equals the work done to stretch or compress the spring, so it equals the elastic potential energy.
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Deriving the equationDeriving the equation
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F
Deriving the equationDeriving the equation
What’s the equation for the force (height) at this position?
x
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Deriving the equationDeriving the equation
kx
x
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kx
Deriving the equationDeriving the equation
x
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Elastic potential energyElastic potential energy
Ep is equal to the work done to
deform the spring by an amount x.
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Elastic potential energyElastic potential energy
This expression is true for more than just springs!
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Elastic potential energyElastic potential energy
Elastic potential energy is stored in all objects that can deform and spring back to their original shape.
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Elastic potential energyElastic potential energy
such as a rubber band . . .
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Typical elastic potential energiesTypical elastic potential energies
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AssessmentAssessment1. What do each of the symbols mean in this equation: Ep = ½ kx2 ?
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AssessmentAssessment1. What do each of the symbols mean in this equation: Ep = ½ kx2 ?
Ep = the elastic potential energy
k = the spring constant in N/mx = the displacement of the end of the spring in meters
2. Translate the equation EP = ½ kx2 into a sentence with the same
meaning.
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AssessmentAssessment1. What do each of the symbols mean in this equation: Ep = ½ kx2 ? Ep =
the elastic potential energyk = the spring constant in N/mx = the displacement of the end of the spring in meters
2. Translate the equation EP = ½ kx2 into a sentence with the same
meaning.The elastic potential energy of a spring is one half the product of its spring constant multiplied by the square of its extension or compression distance.
3. How much elastic potential energy is stored in a 100 N/m spring that is compressed 0.10 meters?
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AssessmentAssessment1. What do each of the symbols mean in this equation: Ep = ½ kx2 ?
Ep = the elastic potential energy
k = the spring constant in N/mx = the displacement of the end of the spring in meters
2. Translate the equation EP = ½ kx2 into a sentence with the same
meaning.The elastic potential energy of a spring is one half the product of its spring constant multiplied by the square of its extension or compression distance.
3. How much elastic potential energy is stored in a 100 N/m spring that is compressed 0.10 meters? 0.50 J
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AssessmentAssessment4. A spring has an elastic potential energy of 100 J when
compressed 0.10 m. What is its spring constant?
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AssessmentAssessment4. A spring has an elastic potential energy of 100 J when
compressed 0.10 m. What is its spring constant?
5. How far is a spring extended if it has 1.0 J of elastic potential energy and its spring constant is 1,000 N/m?
k = 20,000 N/m
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AssessmentAssessment4. A spring has an elastic potential energy of 100 J when
compressed 0.10 m. What is its spring constant?
5. How far is a spring extended if it has 1.0 J of elastic potential energy and its spring constant is 1,000 N/m?
0.045 m or 4.5 cm
k = 20,000 N/m
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AssessmentAssessment6. Are these statements about the spring constant true or false?
a) ___ The spring constant is a measure of the stiffness of the spring.
b) ___ The spring constant tells you how many newtons of force it ……takes to stretch the spring one meter.
c) ___ If a spring stretches easily, it has a high spring constant.
d) ___ The spring constant of a spring varies with x, the amount of …….stretch or compression of the spring.
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a) ___ The spring constant is a measure of the stiffness of the spring.
b) ___ The spring constant tells you how many newtons of force it ……takes to stretch the spring one meter.
c) ___ If a spring stretches easily, it has a high spring constant.
d) ___ The spring constant of a spring varies with x, the amount of …….stretch or compression of the spring.
T
T
F
F
AssessmentAssessment6. Are these statements about the spring constant true or false?