Accelerated frame of reference

Accelerated reference systems are all reference systems that are in accelerated motion compared to an inertial system . This can be an accelerated translational movement and / or an accelerated or non-accelerated rotational movement. An accelerated frame of reference is not an inertial frame.

Although the laws of physics are generally more complicated in accelerated frames of reference (in mechanics, for example, inertia forces must be taken into account when setting up equations of motion ), these frames of reference can in some cases simplify the solution of a problem.

This is usually the case when the reference system is chosen in such a way that the movements are relatively simple:

• Rotating circular or spiral movements around a common center can be z. B. can often be described well when the frame of reference rotates uniformly around the center: The spinning or spiraling body then rests in it or moves along a straight line.

• The Foucault pendulum is usually calculated in a reference system that carries out the rotation of the earth. Likewise, the calculations for the processes in the atmosphere and oceans, on which the forecast of weather and climate development are based.

• Relative movements in a vehicle, e.g. B. that of the wheels are described in a vehicle-mounted system.

• In a reference system that is in free fall in a homogeneous gravity field , the force of gravity is exactly balanced by the force of inertia.

In classical mechanics , time intervals and spatial distances are the same in all reference systems. The conversion of the perceived physical quantities when transitioning to another reference system is therefore carried out by the Euclidean transformation .

kinematics

→ Main article : Kinematics

Inertial system K and accelerated coordinate system K '.

Temporal derivatives in a stationary and a moving reference system

Let P be a point in physical space. In a reference system , it is defined by a position vector that is to be represented with three base vectors ( for the x, y and z directions) and three coordinates : ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {e}}_{i}}$ ${\displaystyle i=1,2,3}$ ${\displaystyle x_{i}}$

${\displaystyle {\vec {r}}\,=\,\sum _{i}x_{i}\,{\vec {e}}_{i}}$

If the point is movable, the coordinates depend on time. ${\displaystyle x_{i}(t)}$

The time derivative of the vector is

${\displaystyle {\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}\,=\,\sum _{i}{\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}{\vec {e}}_{i}}$

It indicates the speed with which the point P moves relative to the reference system. ${\displaystyle {\boldsymbol {K}}}$

Be another frame of reference that moves relative to . Its origin is attached , its base vectors are . Let the position vector of the same point P in K 'be . So that the vectors and define the same physical location in space, the following must apply: ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\vec {R}}(t)}$ ${\displaystyle {\vec {e}}'_{i}(t)}$ ${\displaystyle {\vec {r}}\,'}$ ${\displaystyle {\vec {r}}}$ ${\displaystyle {\vec {r}}\,'}$

${\displaystyle {\vec {r}}\,=\,{\vec {R}}+{\vec {r}}\,'}$.

In the case , the vectors are therefore the same ( ), but their components with respect to or generally not. ${\displaystyle {\vec {R}}={\vec {0}}}$ ${\displaystyle {\vec {r}}={\vec {r}}\,'}$ ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\boldsymbol {K}}'}$

The component representation of in relation to is: ${\displaystyle {\vec {r}}\,'}$ ${\displaystyle {\boldsymbol {K}}'}$

${\displaystyle {\vec {r}}'\,=\,\sum _{i}x_{i}'\,{\vec {e}}_{i}'}$.

The time derivative of the vector relative to the moving system is ${\displaystyle {\vec {r}}\,'}$ ${\displaystyle {\boldsymbol {K}}'}$

${\displaystyle {\frac {\mathrm {d} '{\vec {r}}'}{\mathrm {d} t}}\,=\,\sum _{i}{\frac {\mathrm {d} x_{i}'}{\mathrm {d} t}}{\vec {e}}_{i}'}$

The line in the symbol for the differentiation of a vector means that the coordinates are to be derived that it has in the reference system , so that the derivation denotes a quantity that can be observed there. ${\displaystyle \mathrm {d} '}$ ${\displaystyle {\vec {r}}'(t)}$ ${\displaystyle x_{i}'(t)}$ ${\displaystyle {\boldsymbol {K}}'}$

In order to relate the velocities of the point P as they are observed in and in , the movement of must be described with reference to FIG. As with a rigid body, this movement is at every moment the combination of a translational movement and a rotational movement. The translational motion is given by the speed with which the origin in moved: ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\vec {R}}(t)}$ ${\displaystyle {\boldsymbol {K}}}$

${\displaystyle {\vec {v}}_{trans}(t)\,=\,{\frac {\mathrm {d} {\vec {R}}}{\mathrm {d} t}}}$.

Due to the translational movement, all points with a constant position vector move in parallel, so the base vectors also remain constant over time. However, due to the rotational movement, these change. The current rotational movement of has an axis of rotation through the origin at the location and an angular velocity , which are combined with the direction of rotation to form the vectorial angular velocity . The basis vectors change from in with the speed (see path speed ): ${\displaystyle {\vec {r}}\,'}$ ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\vec {e}}'_{i}}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\vec {R}}(t)}$ ${\displaystyle \omega (t)}$ ${\displaystyle {\vec {\omega }}(t)}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\boldsymbol {K}}}$

${\displaystyle {\frac {\mathrm {d} {\vec {e}}'_{i}}{\mathrm {d} t}}\,=\,{\vec {\omega }}\times {\vec {e}}'_{i}}$

This allows the time derivative of the vector as it appears in the reference system to be calculated. According to the product rule is ${\displaystyle {\vec {r}}'(t)}$ ${\displaystyle {\boldsymbol {K}}}$

${\displaystyle {\frac {\mathrm {d} {\vec {r}}\,'}{\mathrm {d} t}}\,=\,\sum _{i}{\frac {\mathrm {d} x'_{i}}{\mathrm {d} t}}{\vec {e}}'_{i}+\sum _{i}x'_{i}{\frac {\mathrm {d} {\vec {e}}'_{i}}{\mathrm {d} t}}}$.

According to the above formulas, this is the same as

${\displaystyle {\frac {\mathrm {d} {\vec {r}}\,'}{\mathrm {d} t}}\,=\,{\frac {\mathrm {d} '{\vec {r}}\,'}{\mathrm {d} t}}\,+\,\sum _{i}x'_{i}\,({\vec {\omega }}\times {\vec {e}}'_{i})\,=\,{\frac {\mathrm {d} '{\vec {r}}\,'}{\mathrm {d} t}}+{\vec {\omega }}\times {\vec {r}}\,'}$.

This formula is often abbreviated as an operator equation

${\displaystyle {\frac {\mathrm {d} \bullet }{\mathrm {d} t}}={\frac {\mathrm {d} '\bullet }{\mathrm {d} t}}+{\vec {\omega }}\times \bullet }$.

Applied to any vector (to be used in ), it provides the relationship between its rate of change or in . ${\displaystyle \bullet }$ ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\boldsymbol {K}}'}$

Transformation of speed

, In the following based on the technical mechanics , which in the reference system observed variables as absolute speed and absolute acceleration referred, and on related variables as relative velocity and relative acceleration . ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\boldsymbol {K}}'}$

The absolute speed of the point is: ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {v}}\,=\,{\frac {\mathrm {d} {\vec {r}}}{\mathrm {d} t}}\,=\,\sum _{i}{\frac {\mathrm {d} x_{i}}{\mathrm {d} t}}{\vec {e}}_{i}\left(\,=\,\sum _{i}{\dot {x}}_{i}{\vec {e}}_{i}\right)}$

The relative speed of the point is analogous: ${\displaystyle {\vec {v}}'}$

${\displaystyle {\vec {v}}'\,=\,{\frac {\mathrm {d} '{\vec {r}}'}{\mathrm {d} t}}\,=\,\sum _{i}{\frac {\mathrm {d} x_{i}'}{\mathrm {d} t}}{\vec {e}}_{i}'}$

Because of the absolute speed : ${\displaystyle {\vec {r}}\,=\,{\vec {R}}+{\vec {r}}\,'}$ ${\displaystyle {\vec {v}}}$

${\displaystyle {\vec {v}}\,\,=\,\,{\frac {\mathrm {d} }{\mathrm {d} t}}({\vec {R}}+{\vec {r}}')\,=\,{\frac {\mathrm {d} {\vec {R}}}{\mathrm {d} t}}\,+\,{\frac {\mathrm {d} '{\vec {r}}'}{\mathrm {d} t}}\,+\,{\vec {\omega }}\times {\vec {r}}'\,=\,{\vec {v}}_{trans}\,+\,{\vec {\omega }}\times {\vec {r}}'\,+\,{\vec {v}}\,'}$.

The part ( ) of the absolute speed is called the guide speed. All points that are at rest in the reference system move in the reference system with the guide speed. If they do not rest, their relative speed must be added to the guidance speed . ${\displaystyle {\vec {v}}_{trans}+{\vec {\omega }}\times {\vec {r}}'}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\vec {v}}\,'}$

Transformation of acceleration

The time derivative of the formula for the speed of the point P in gives the absolute acceleration, expressed by the observable quantities and and : ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\vec {r}}'}$ ${\displaystyle {\vec {v}}'}$ ${\displaystyle {\vec {a}}'}$

${\displaystyle {\frac {\mathrm {d} {\vec {v}}}{\mathrm {d} t}}\,=\,{\frac {\mathrm {d} }{\mathrm {d} t}}\left({\vec {v}}_{trans}\,+\,{\vec {\omega }}\times {\vec {r}}'\,+\,{\vec {v}}'\right)\,=\,\underbrace {\frac {\mathrm {d} {\vec {v}}_{trans}}{\mathrm {d} t}} _{{\vec {a}}_{trans}}\,+\,\underbrace {\left({\frac {\mathrm {d} {\vec {\omega }}}{\mathrm {d} t}}\right)} _{\dot {\vec {\omega }}}\times {\vec {r}}'\,+\,{\vec {\omega }}\times \underbrace {\left({\frac {\mathrm {d} {\vec {r}}\,'}{\mathrm {d} t}}\right)} _{{\vec {v}}\,'+\,{\vec {\omega }}\times {\vec {r}}\,'}\,+\,\underbrace {\frac {\mathrm {d} {\vec {v}}'}{\mathrm {d} t}} _{{\vec {a}}\,'+\,{\vec {\omega }}\times {\vec {v}}\,'}}$

The above operator equation must be applied to and once . The sizes are inserted according to the above formulas and rearranged somewhat: ${\displaystyle {\vec {r}}'}$ ${\displaystyle {\vec {v}}'}$

${\displaystyle {\vec {a}}\,=\,{\vec {a}}_{trans}\,+\,{\dot {\vec {\omega }}}\times {\vec {r}}'\,+{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}\,')\,+\,{\vec {a}}'\,+2\,{\vec {\omega }}\times {\vec {v}}\,'}$

The accelerations and , in or differ not only in terms of the translational acceleration of the system in the system . The three additional summands are: ${\displaystyle {\vec {a}}}$ ${\displaystyle {\vec {a}}'}$ ${\displaystyle {\boldsymbol {K}}}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\vec {a}}_{trans}}$ ${\displaystyle {\boldsymbol {K}}'}$ ${\displaystyle {\boldsymbol {K}}'}$

${\displaystyle {\vec {a}}_{Euler}\,=\,-{\dot {\vec {\omega }}}\times {\vec {r}}'}$	Euler acceleration in ${\displaystyle {\boldsymbol {K}}'}$
${\displaystyle {\vec {a}}_{Zentrifugal}\,=\,-{\vec {\omega }}\times ({\vec {\omega }}\times {\vec {r}}\,')}$	Centrifugal acceleration in ${\displaystyle {\boldsymbol {K}}'}$
${\displaystyle {\vec {a}}_{Coriolis}\,=\,-2\,{\vec {\omega }}\times {\vec {v}}\,'}$	Coriolis acceleration in ${\displaystyle {\boldsymbol {K}}'}$

(Note: is sometimes also defined with the opposite sign.) ${\displaystyle {\vec {a}}_{Coriolis}}$

In this definition, the acceleration in the inertial system is the sum of the guide acceleration, the relative acceleration and the Coriolis acceleration, the guide acceleration being the acceleration that a body has when it is permanently connected to the coordinate system: ${\displaystyle {\vec {a}}_{F}}$ ${\displaystyle {\vec {a}}'}$

${\displaystyle {\vec {a}}={\vec {a}}_{F}+{\vec {a}}'+{\vec {a}}_{Coriolis}}$

The result shows that if a point is at rest in one frame of reference or moves in a straight line, it generally has not only a different speed but also a different acceleration in another moving frame of reference. The differences in the observed accelerations are interpreted as the effect of inertial forces . More see there.

See also

• Orthogonal tensor # Rigid body motions

literature

• F. Scheck: Theoretical Physics 1. Mechanics . Springer Verlag, ISBN 978-3-540-71377-7
• Jürgen Dankert and Helga Dankert: Technical Mechanics. Springer, 6th edition, 2011, chap. 27.2 ff.
• Martin Mayr: Technical mechanics: statics, kinematics - kinetics - vibrations, strength theory . 6th revised edition. Hanser, 2008, ISBN 978-3-446-41690-1 ( limited preview in Google book search).

References and comments

1. ↑ K. Marguerre: Technical Mechanics: Third Part: Kinetics . Springer-Verlag, 1968, ISBN 978-3-540-04173-3 , pp. 67 . : ( limited preview in Google Book search)
2. ↑ This choice of words does not mean that there is such a thing as "absolute calm" or "absolute speed" in classical mechanics. See Principle of Relativity # Classical Mechanics .