Rounding and rotation

Содержание и основные понятия

Rotation is convenient to describe angle, angle speed and angular acceleration. The linear characteristics of the point depend on its distance to the axis.
GEF · 10 grade
PIF · Baseline
Interactive model
angular speed
angular acceleration
Rotation
moment
01
The physical picture
What happens and what boundaries the model has.
Main message
Rotation is convenient to describe angle, angle speed and angular acceleration. The linear characteristics of the point depend on its distance to the axis.
All points of the solid body have the same angular velocity, but different linear speeds.
The normal acceleration is directed towards the axle, the tangential is related to the change of the speed module.
The moment of force determines the force's ability to rotate.
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.
v
=
ω
R
v=\omega R
v
=
ω
R
Linear speed
a
ц
=
ω
2
R
a_{\mathrm{ц}}=\omega^2R
a
c/
​
=
ω
2
R
normal acceleration
M
=
F
l
M=Fl
M
=
Fl
force moment
Review of the situation
The edge of the disc moves faster than the dot at the axis, although both turn in the same time.
04
Links and application
In the real world
The edge of the disc moves faster than the dot at the axis, although both turn in the same time.
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 theme "Cycle motion and rotation" through the model, not just through the formula.
Outcome 2
Distinguish and correctly use terms: angular velocity, angular acceleration, rotation.
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.
angular speed
Production angle of rotation over time; characteristic of speed of rotation.
angular acceleration
It's a time-to-time linear velocity.
Rotation
The movement of the body at which its dots describe the circles around the common axis.
moment
A value describing the rotating effect of the force or the distribution of the physical value in relation to the point or axis.
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.
v
=
ω
R
v=\omega R
v
=
ω
R
model effect
Linear speed.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
s
=
R
φ
s=R\varphi
s
=
Rφ
The length of the arc of the circle is equal to a product of radius to the angle in the radians.
v
=
Δ
s
Δ
t
=
R
Δ
φ
Δ
t
v=\frac{\Delta s}{\Delta t}=R\frac{\Delta\varphi}{\Delta t}
v
=
Δ
t
Δ
s
​
=
R
Δ
t
Δ
φ
​
They divide the arc over time and use the constant radius.
ω
=
Δ
φ
Δ
t
⇒
v
=
ω
R
\omega=\frac{\Delta\varphi}{\Delta t}\quad\Rightarrow\quad v=\omega R
ω
=
Δ
t
Δ
φ
​
⇒
v
=
ω
R
The angle-to-time ratio is the angular velocity.
Conditions:
The point shall be rotated along the circle of the radius R; for instantaneous values the limit shall be used and v means the relative speed module.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
a
ц
=
ω
2
R
a_{\mathrm{ц}}=\omega^2R
a
c/
​
=
ω
2
R
model effect
Normal acceleration.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
a
ц
=
v
2
R
a_{\mathrm{ц}}=\frac{v^2}{R}
a
c/
​
=
R
v
2
​
A normal acceleration v2/R shall be obtained from the change in direction of the speed vector on the circle.
v
=
ω
R
v=\omega R
v
=
ω
R
The linear and angular velocity of the point is bound by the radius of the trajectory.
a
ц
=
(
ω
R
)
2
R
=
ω
2
R
a_{\mathrm{ц}}=\frac{(\omega R)^2}{R}=\omega^2R
a
c/
​
=
R
(
ω
R
)
2
​
=
ω
2
R
The setting and abbreviation of one R factor give the required shape.
Conditions:
The motion along the circle of the radius R; the formula sets the acceleration component module to the center and is also valid for the changing . . instantaneous.
Monitoring:
The dimensions of both parts shall coincide; the result shall be checked for sign, order and limit case.
M
=
F
l
M=Fl
M
=
Fl
definition
A moment of strength.
The following is an indication of the definition, experience or law from which this entry arises and where it is applicable.
l
=
R
sin
⁡
α
l=R\sin\alpha
l
=
R
sin
α
The shoulder of force is the shortest distance from the axis to the force line.
M
≡
F
R
sin
⁡
α
M\equiv FR\sin\alpha
M
≡
FR
sin
α
The moment module is defined as the vector work module of the radius vector on force.
M
=
F
l
M=Fl
M
=
Fl
The replacement of R sine α with a shoulder l gives a school scalar record.
Conditions:
M = Fl sets the moment module relative to the point or moment relative to an axis perpendicular to the plane of the vectors r and F; l is the perpendicular distance from the point to the force line. For the arbitrary axis, the projection of the victor r x F on this axis shall be used.
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 key condition of this subject is that all points of the solid body have the same angular velocity, but different linear speeds.
How to Check the Mind
The normal acceleration is directed towards the axle, the tangential is related to the change of the speed module. Then check the units, vector direction and behaviour of the result at zero or very large parameter.
Σ
Conclusion on the topic
Rounding and rotation
Rotation is convenient to describe angle, angle speed and angular acceleration. The linear characteristics of the point depend on its distance to the axis.
All points of the solid body have the same angular velocity, but different linear speeds.
The normal acceleration is directed towards the axle, the tangential is related to the change of the speed module.
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.