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| Meta Title | Hooke's law |
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| Boilerpipe Text | Next:
Motion in a general
Up:
Conservation of energy
Previous:
Potential energy
Consider a mass
which slides over a horizontal frictionless surface. Suppose that
the mass is attached
to a light horizontal spring whose other end is anchored to an immovable object. See
Fig.
42
. Let
be the extension of the spring:
i.e.
, the difference between
the spring's actual length and its unstretched length. Obviously,
can also be used as
a coordinate to determine the horizontal displacement of the mass.
According to Hooke's law, the force
that the spring exerts on the mass is directly proportional to its extension, and always acts to reduce
this extension.
Hence, we can write
(159)
where the positive quantity
is called the
force constant
of the spring. Note that
the minus sign in the above equation ensures that the force always acts to reduce the
spring's extension:
e.g.
, if the extension is positive then the force acts to the left, so as to
shorten the spring.
Figure 42:
Mass on a spring
According to Eq. (
140
), the work performed by the spring force on the mass
as it moves from displacement
to
is
(160)
Note that the right-hand side of the above expression consists of the difference
between two factors: the first only depends on the final state of the mass, whereas the
second only depends on its initial state. This is a sure sign that it is
possible to associate a
potential energy
with the spring force.
Equation (
155
), which is the basic definition of potential
energy, yields
(161)
Hence, the potential energy of the mass takes the form
(162)
Note that the above potential energy actually represents energy stored
by the
spring
--in
the form of mechanical stresses--when it is either
stretched or compressed. Incidentally, this energy must be stored
without loss
, otherwise
the concept of potential energy would be meaningless. It follows that the spring
force is another example of a conservative force.
It is reasonable to suppose that the form of the spring potential energy is somehow related to the
form of the spring force. Let us now explicitly investigate this
relationship. If we let
and
then Eq. (
161
)
gives
(163)
We can differentiate this expression to obtain
(164)
Thus, in 1-dimension, a conservative force is equal to minus the derivative (with respect
to displacement) of its associated potential energy. This is a quite general result.
For the case of a
spring force:
, so
.
As is easily demonstrated, the 3-dimensional equivalent to Eq. (
164
) is
(165)
For example, we have seen that the gravitational potential energy of
a mass
moving above the Earth's surface is
,
where
measures height off the ground. It follows that the
associated gravitational force is
(166)
In other words, the force is of magnitude
, and is directed
vertically downward.
The total energy of the mass shown in Fig.
42
is the sum of its kinetic and
potential energies:
(167)
Of course,
remains constant during the
mass's motion. Hence, the above expression can be rearranged
to give
(168)
Since it is impossible
for a kinetic energy
to be negative, the above expression
suggests that
can never exceed the value
(169)
Here,
is termed the
amplitude
of the mass's motion. Note that when
attains its maximum value
, or its minimum value
, the
kinetic energy is momentarily zero (
i.e.
,
).
Next:
Motion in a general
Up:
Conservation of energy
Previous:
Potential energy
Richard Fitzpatrick
2006-02-02 |
| Markdown | [](https://farside.ph.utexas.edu/teaching/301/lectures/node62.html) [](https://farside.ph.utexas.edu/teaching/301/lectures/node55.html) [](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html)
**Next:** [Motion in a general](https://farside.ph.utexas.edu/teaching/301/lectures/node62.html) **Up:** [Conservation of energy](https://farside.ph.utexas.edu/teaching/301/lectures/node55.html) **Previous:** [Potential energy](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html)
## Hooke's law
Consider a mass  which slides over a horizontal frictionless surface. Suppose that the mass is attached to a light horizontal spring whose other end is anchored to an immovable object. See Fig. [42](https://farside.ph.utexas.edu/teaching/301/lectures/node61.html#f42). Let  be the extension of the spring: *i.e.*, the difference between the spring's actual length and its unstretched length. Obviously,  can also be used as a coordinate to determine the horizontal displacement of the mass. According to Hooke's law, the force  that the spring exerts on the mass is directly proportional to its extension, and always acts to reduce this extension. Hence, we can write
| | |
|---|---|
|  | (159) |
where the positive quantity  is called the *force constant* of the spring. Note that the minus sign in the above equation ensures that the force always acts to reduce the spring's extension: *e.g.*, if the extension is positive then the force acts to the left, so as to shorten the spring.
| |
|---|
|  |
According to Eq. ([140](https://farside.ph.utexas.edu/teaching/301/lectures/node58.html#eaax)), the work performed by the spring force on the mass as it moves from displacement  to  is
| | |
|---|---|
| ![\\begin{displaymath} W = \\int\_{x\_A}^{x\_B} f(x) dx = -k\\int\_{x\_A}^{x\_B} x dx = -... ... \\frac{1}{2} k x\_B^{ 2} -\\frac{1}{2} k x\_A^{ 2} \\right\]. \\end{displaymath}](https://farside.ph.utexas.edu/teaching/301/lectures/img674.png) | (160) |
Note that the right-hand side of the above expression consists of the difference between two factors: the first only depends on the final state of the mass, whereas the second only depends on its initial state. This is a sure sign that it is possible to associate a *potential energy* with the spring force. Equation ([155](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html#epot)), which is the basic definition of potential energy, yields
| | |
|---|---|
|  | (161) |
Hence, the potential energy of the mass takes the form
| | |
|---|---|
|  | (162) |
Note that the above potential energy actually represents energy stored *by the spring*\--in the form of mechanical stresses--when it is either stretched or compressed. Incidentally, this energy must be stored *without loss*, otherwise the concept of potential energy would be meaningless. It follows that the spring force is another example of a conservative force.
It is reasonable to suppose that the form of the spring potential energy is somehow related to the form of the spring force. Let us now explicitly investigate this relationship. If we let  and  then Eq. ([161](https://farside.ph.utexas.edu/teaching/301/lectures/node61.html#epotn)) gives
| | |
|---|---|
|  | (163) |
We can differentiate this expression to obtain
| | |
|---|---|
|  | (164) |
Thus, in 1-dimension, a conservative force is equal to minus the derivative (with respect to displacement) of its associated potential energy. This is a quite general result. For the case of a spring force: , so .
As is easily demonstrated, the 3-dimensional equivalent to Eq. ([164](https://farside.ph.utexas.edu/teaching/301/lectures/node61.html#egrad)) is
| | |
|---|---|
|  | (165) |
For example, we have seen that the gravitational potential energy of a mass  moving above the Earth's surface is , where  measures height off the ground. It follows that the associated gravitational force is
| | |
|---|---|
|  | (166) |
In other words, the force is of magnitude , and is directed vertically downward.
The total energy of the mass shown in Fig. [42](https://farside.ph.utexas.edu/teaching/301/lectures/node61.html#f42) is the sum of its kinetic and potential energies:
| | |
|---|---|
|  | (167) |
Of course,  remains constant during the mass's motion. Hence, the above expression can be rearranged to give
| | |
|---|---|
|  | (168) |
Since it is impossible for a kinetic energy to be negative, the above expression suggests that  can never exceed the value
| | |
|---|---|
|  | (169) |
Here,  is termed the *amplitude* of the mass's motion. Note that when  attains its maximum value , or its minimum value , the kinetic energy is momentarily zero (*i.e.*, ).
***
[](https://farside.ph.utexas.edu/teaching/301/lectures/node62.html) [](https://farside.ph.utexas.edu/teaching/301/lectures/node55.html) [](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html)
**Next:** [Motion in a general](https://farside.ph.utexas.edu/teaching/301/lectures/node62.html) **Up:** [Conservation of energy](https://farside.ph.utexas.edu/teaching/301/lectures/node55.html) **Previous:** [Potential energy](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html)
Richard Fitzpatrick 2006-02-02 |
| Readable Markdown | [](https://farside.ph.utexas.edu/teaching/301/lectures/node62.html) [](https://farside.ph.utexas.edu/teaching/301/lectures/node55.html) [](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html)
**Next:** [Motion in a general](https://farside.ph.utexas.edu/teaching/301/lectures/node62.html) **Up:** [Conservation of energy](https://farside.ph.utexas.edu/teaching/301/lectures/node55.html) **Previous:** [Potential energy](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html)
Consider a mass  which slides over a horizontal frictionless surface. Suppose that the mass is attached to a light horizontal spring whose other end is anchored to an immovable object. See Fig. [42](https://farside.ph.utexas.edu/teaching/301/lectures/node61.html#f42). Let  be the extension of the spring: *i.e.*, the difference between the spring's actual length and its unstretched length. Obviously,  can also be used as a coordinate to determine the horizontal displacement of the mass. According to Hooke's law, the force  that the spring exerts on the mass is directly proportional to its extension, and always acts to reduce this extension. Hence, we can write
| | |
|---|---|
|  | (159) |
where the positive quantity  is called the *force constant* of the spring. Note that the minus sign in the above equation ensures that the force always acts to reduce the spring's extension: *e.g.*, if the extension is positive then the force acts to the left, so as to shorten the spring.
| |
|---|
|  |
According to Eq. ([140](https://farside.ph.utexas.edu/teaching/301/lectures/node58.html#eaax)), the work performed by the spring force on the mass as it moves from displacement  to  is
| | |
|---|---|
| ![\\begin{displaymath} W = \\int\_{x\_A}^{x\_B} f(x) dx = -k\\int\_{x\_A}^{x\_B} x dx = -... ... \\frac{1}{2} k x\_B^{ 2} -\\frac{1}{2} k x\_A^{ 2} \\right\]. \\end{displaymath}](https://farside.ph.utexas.edu/teaching/301/lectures/img674.png) | (160) |
Note that the right-hand side of the above expression consists of the difference between two factors: the first only depends on the final state of the mass, whereas the second only depends on its initial state. This is a sure sign that it is possible to associate a *potential energy* with the spring force. Equation ([155](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html#epot)), which is the basic definition of potential energy, yields
| | |
|---|---|
|  | (161) |
Hence, the potential energy of the mass takes the form
| | |
|---|---|
|  | (162) |
Note that the above potential energy actually represents energy stored *by the spring*\--in the form of mechanical stresses--when it is either stretched or compressed. Incidentally, this energy must be stored *without loss*, otherwise the concept of potential energy would be meaningless. It follows that the spring force is another example of a conservative force.
It is reasonable to suppose that the form of the spring potential energy is somehow related to the form of the spring force. Let us now explicitly investigate this relationship. If we let  and  then Eq. ([161](https://farside.ph.utexas.edu/teaching/301/lectures/node61.html#epotn)) gives
| | |
|---|---|
|  | (163) |
We can differentiate this expression to obtain
| | |
|---|---|
|  | (164) |
Thus, in 1-dimension, a conservative force is equal to minus the derivative (with respect to displacement) of its associated potential energy. This is a quite general result. For the case of a spring force: , so .
As is easily demonstrated, the 3-dimensional equivalent to Eq. ([164](https://farside.ph.utexas.edu/teaching/301/lectures/node61.html#egrad)) is
| | |
|---|---|
|  | (165) |
For example, we have seen that the gravitational potential energy of a mass  moving above the Earth's surface is , where  measures height off the ground. It follows that the associated gravitational force is
| | |
|---|---|
|  | (166) |
In other words, the force is of magnitude , and is directed vertically downward.
The total energy of the mass shown in Fig. [42](https://farside.ph.utexas.edu/teaching/301/lectures/node61.html#f42) is the sum of its kinetic and potential energies:
| | |
|---|---|
|  | (167) |
Of course,  remains constant during the mass's motion. Hence, the above expression can be rearranged to give
| | |
|---|---|
|  | (168) |
Since it is impossible for a kinetic energy to be negative, the above expression suggests that  can never exceed the value
| | |
|---|---|
|  | (169) |
Here,  is termed the *amplitude* of the mass's motion. Note that when  attains its maximum value , or its minimum value , the kinetic energy is momentarily zero (*i.e.*, ).
***
[](https://farside.ph.utexas.edu/teaching/301/lectures/node62.html) [](https://farside.ph.utexas.edu/teaching/301/lectures/node55.html) [](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html)
**Next:** [Motion in a general](https://farside.ph.utexas.edu/teaching/301/lectures/node62.html) **Up:** [Conservation of energy](https://farside.ph.utexas.edu/teaching/301/lectures/node55.html) **Previous:** [Potential energy](https://farside.ph.utexas.edu/teaching/301/lectures/node60.html)
Richard Fitzpatrick 2006-02-02 |
| Shard | 65 (laksa) |
| Root Hash | 16797036158535431465 |
| Unparsed URL | edu,utexas!ph,farside,/teaching/301/lectures/node61.html s443 |