Work and potential fields

Содержание и основные понятия

The work of the force is equal to the integral measure of the transmission of energy during displacement. For potential forces, the work is independent of the path and equal to loss of the potential energy.
GEF · 10 grade
PIF · Baseline
Interactive model
work
Power
Potential field
dissipation
01
The physical picture
What happens and what boundaries the model has.
Main message
The work of the force is equal to the integral measure of the transmission of energy during displacement. For potential forces, the work is independent of the path and equal to loss of the potential energy.
The variable force work corresponds to the area under schedule F(s).
Full mechanical energy is maintained if only potential forces are active.
Dissipative forces convert orderly mechanical energy into internal energy.
How to reason
First, name the system and the interaction, then select the values, and only then write down the law. The formula works within the model's assumptions.
02
Laws and values
Symbols should be read as a relationship of values, not as a recipe for substitution.
A
рез
=
Δ
E
к
A_{\mathrm{рез}}=\Delta E_{\mathrm{к}}
A
res.
​
=
Δ
E
to
​
Theorem of kinetic energy
A
пот
=
−
Δ
E
п
A_{\mathrm{пот}}=-\Delta E_{\mathrm{п}}
A
sweat.
​
=
−
Δ
E
c/
​
Potential force work
Review of the situation
When the spring is stretched, the external force works as kx2/2 if the loss is negligible.
04
Links and application
In the real world
When the spring is stretched, the external force works as kx2/2 if the loss is negligible.
Model verification
Compare the prediction with observation, evaluate the units, the order of magnitude and the influence of the factors not taken into account. Disparity does not always mean an error of law: often idealization is no longer sufficient.
C
What needs to be understood
Outcome 1
Explain the physical meaning of the topic "Work and potential fields" through the model, not just through the formula.
Outcome 2
Identify and correctly use terms: work, power, potential field.
Outcome 3
Indicate the conditions for the applicability of the law, verify units and limit cases.
T
Terms of the topic
The definition should indicate the physical meaning, not a substitute for a domestic synonym.
work
The scalar measure of energy transmission by force when moving the point of its application.
Power
Work or energy transferred per unit of time.
Potential field
A field whose forces do not depend on the shape of the path and is equal to minus the change of potential energy.
dissipation
Irreversible transformation of orderly mechanical energy into internal energy.
F
Origin and conclusion of the formulas
The fundamental law cannot be "proven" by algebras: for each record, it is stated whether it is a definition, an experimental law, or a model effect.
A
рез
=
Δ
E
к
A_{\mathrm{рез}}=\Delta E_{\mathrm{к}}
A
res.
​
=
Δ
E
to
​
model effect
The theory of kinetic energy.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
d
A
=
∑
F
⃗
⋅
d
r
⃗
dA=\sum\vec F\cdot d\vec r
d
A
=
∑
F
⋅
d
r
Elementary work equals its scalar work to elementary movement.
d
A
=
m
d
v
⃗
d
t
⋅
v
⃗
d
t
=
m
v
⃗
⋅
d
v
⃗
dA=m\frac{d\vec v}{dt}\cdot\vec v\,dt=m\vec v\cdot d\vec v
d
A
=
m
d
t
d
v
​
⋅
v
d
t
=
m
v
⋅
d
v
The addition of the second law of Newton and dr = v dt connects the work to the change of speed.
d
A
=
d
⁣
(
m
v
2
2
)
⇒
A
1
→
2
=
m
v
2
2
2
−
m
v
1
2
2
=
Δ
E
к
dA=d\!\left(\frac{mv^2}{2}\right)\Rightarrow A_{1\to2}=\frac{mv_2^2}{2}-\frac{mv_1^2}{2}=\Delta E_{\mathrm{к}}
d
A
=
d
(
2
m
v
2
​
)
⇒
A
1
→
2
​
=
2
m
v
2
2
​
​
−
2
m
v
1
2
​
​
=
Δ
E
to
​
Integration provides the theorem for arbitrary trajectory and a variable equal to constant mass.
Conditions:
The physical point of constant mass in the inertial system; A is the combined operation of all forces; for variable forces, the work is calculated by integration.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
A
пот
=
−
Δ
E
п
A_{\mathrm{пот}}=-\Delta E_{\mathrm{п}}
A
sweat.
​
=
−
Δ
E
c/
​
definition
The work of potential power.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
E
п
,
2
−
E
п
,
1
≡
−
A
1
→
2
(
пот
)
E_{\mathrm{п},2}-E_{\mathrm{п},1}\equiv-A_{1\to2}^{(\mathrm{пот})}
E
c/
,
2
​
−
E
c/
,
1
​
≡
−
A
1
→
2
(
sweat.
)
​
The change in potential energy is defined as the operation of a potential force with the opposite sign.
Δ
E
п
=
E
п
,
2
−
E
п
,
1
\Delta E_{\mathrm{п}}=E_{\mathrm{п},2}-E_{\mathrm{п},1}
Δ
E
c/
​
=
E
c/
,
2
​
−
E
c/
,
1
​
Write down the standard energy change symbol between the two positions.
A
пот
=
−
Δ
E
п
A_{\mathrm{пот}}=-\Delta E_{\mathrm{п}}
A
sweat.
​
=
−
Δ
E
c/
​
The rewiring of the parts of the definition gives the formula for the operation of the potential force.
Conditions:
Only for a conservative force that does not depend on the path; the zero level of potential energy is chosen at random.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
!
Model boundaries and typical errors
Error: Forget model condition
The formula cannot be transferred to any situation simply because of the overlap of letters. The key condition of this topic is that the variable force work corresponds to the area under the F(s) schedule.
How to Check the Mind
Full mechanical energy is stored if only potential forces are active. Then check the units, vector direction and result behaviour at zero or very large parameter.
Σ
Conclusion on the topic
Work and potential fields
The work of the force is equal to the integral measure of the transmission of energy during displacement. For potential forces, the work is independent of the path and equal to loss of the potential energy.
The variable force work corresponds to the area under schedule F(s).
Full mechanical energy is maintained if only potential forces are active.
Any numerical result shall be accompanied by a unit, a sign and an explanation of its physical meaning.
The content is consistent with the FRP on Physics of the OS/OSS 2025 and the subject matter of the current lines of the PNP: I. M. Peryškin and co-authors (7–9), G. J. Mäkishev and coauthors (10–11), and with the lines "Engineers of the Future" and V. A. Kasyanov. The original texts are not reproduced, the text of the textbooks is not reproduced.