Balance of the solid body

Содержание и основные понятия

For the equilibrium of steady and rotating motion, the sum of the forces and moments relative to any axle shall be equal to zero at the same time.
GEF · 10 grade
PIF · Baseline
Interactive model
force moment
Shoulder
centre of masses
Sustainability
01
The physical picture
What happens and what boundaries the model has.
Main message
For the equilibrium of steady and rotating motion, the sum of the forces and moments relative to any axle shall be equal to zero at the same time.
The moment of strength depends on the force and the perpendicular shoulder.
The selection of the axis through unknown reactions often simplifys the equation of moments.
Sustainability depends on the position of the centre of the masses and the area of the support.
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.
∑
F
⃗
=
0
⃗
\sum\vec F=\vec 0
∑
F
=
0
Balance of power
∑
M
⃗
O
=
0
⃗
\sum\vec M_O=\vec 0
∑
M
O
​
=
0
balance of moments
Review of the situation
The uniform beam in equilibrium has the centre of gravity in the middle, and the reactions of the pillars are found from the balance of forces and moments.
04
Links and application
In the real world
The uniform beam in equilibrium has the centre of gravity in the middle, and the reactions of the pillars are found from the balance of forces and moments.
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 "Equivalence of the solid body" through the model, not just the formula.
Outcome 2
Distinguish and correctly use terms: moment of strength, shoulder, center of mass.
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.
force moment
Measure of rotation of force relative to the axis or point.
Shoulder
The maximum distance from the axis of rotation to the force line.
centre of masses
The point whose motion describes the progress of the system as a whole.
Sustainability
The ability of the equilibrium system to return to its original state after a slight deviation.
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.
∑
F
⃗
=
0
⃗
\sum\vec F=\vec 0
∑
F
=
0
model effect
Balance of power.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
∑
F
⃗
=
m
a
⃗
цм
\sum\vec F=m\vec a_{\mathrm{цм}}
∑
F
=
m
a
zm
​
The continuous movement of the center of masses is described in Newton's second law.
a
⃗
цм
=
0
⃗
\vec a_{\mathrm{цм}}=\vec 0
a
zm
​
=
0
The centre of the masses settles or moves evenly when the equilibrium is steady.
∑
F
⃗
=
0
⃗
\sum\vec F=\vec 0
∑
F
=
0
Therefore, the equidistant external force must be zero.
Conditions:
Inertial counting system; this is a condition for the gradual equilibrium of the solid body, which is applied together with .M = 0 for full equilibrium.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
∑
M
⃗
O
=
0
⃗
\sum\vec M_O=\vec 0
∑
M
O
​
=
0
model effect
The balance of moments.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
∑
M
⃗
O
=
d
L
⃗
O
d
t
\sum\vec M_O=\frac{d\vec L_O}{dt}
∑
M
O
​
=
d
t
d
L
O
​
​
The sum of the moments of the external forces relative to the stationary point of the inertial system is equal to the rate of change of the pulse moment.
d
L
⃗
O
d
t
=
0
⃗
\frac{d\vec L_O}{dt}=\vec 0
d
t
d
L
O
​
​
=
0
The moment of body impulse does not change when there is a rotating equilibrium.
∑
M
⃗
O
=
0
⃗
\sum\vec M_O=\vec 0
∑
M
O
​
=
0
Therefore, the vector sum of the moments of the external forces is zero; in a flat task, this becomes the algebraic sum M_O = 0.
Conditions:
A solid body in an inertial system; the moments of all external forces are taken relative to one fixed point. The flat task uses the projection of moments on one axis and the single rule of the signs.
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 coincidence of the letters. The key condition of this topic is that the moment of force depends on the force and the perpendicular shoulder.
How to Check the Mind
The selection of the axis through unknown reactions often simplifys the moment equation. Then check the unit, vector direction and result behaviour at zero or very large parameter.
Σ
Conclusion on the topic
Balance of the solid body
For the equilibrium of steady and rotating motion, the sum of the forces and moments relative to any axle shall be equal to zero at the same time.
The moment of strength depends on the force and the perpendicular shoulder.
The selection of the axis through unknown reactions often simplifys the equation of moments.
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.