Name: dimensional checks for physics equations
Uploaded: 2023-05-16T08:18:40.000Z
Duration: 1 min 8 s
Channel: JoVE

Nehmen wir an, ein einfaches Pendel der Masse m ist an einer Saite der Länge L befestigt, die unter dem Einfluss der Schwerkraft g schwingt. Wie sieht die Gleichung für die Zeitspanne des Pendels aus?
Identifizieren und listen Sie zunächst die Variablen auf, die an dem Problem beteiligt sind. Die Zeitspanne T kann als Produkt dieser Variablen ausgedrückt werden, die jeweils auf einen unbekannten Exponenten angehoben werden. Hier ist k eine dimensionslose Konstante.
Unter Ausschluss der dimensionslosen Konstante erhält man eine Gleichung, die die Dimensionen der Variablen mit dem Zeitraum in Beziehung setzt.
Indem man nun die Exponenten der Dimensionen auf beiden Seiten gleichsetzt und die Gleichungen löst, werden die Werte der unbekannten Exponenten bestimmt.
Beim Einsetzen der Exponenten erhält man den endgültigen Ausdruck für die Zeitspanne, der ein Produkt aus der Konstanten k und der Quadratwurzel der Länge über der Gravitationsbeschleunigung ist.
Eine der Einschränkungen der dimensionalen Analyse besteht darin, dass sie es uns nicht erlaubt, den Wert der dimensionslosen Konstante k zu finden.

<ol>
	<li>OpenStax. (2019). Universitätsphysik Vol. 1. [Webversion]. Abgerufen von <a href="https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis">https://openstax.org/books/university-physics-volume-1/pages/1-4-dimensional-analysis</a>; Abschnitt 1.4; Seiten 22-25.</li>
</ol>

problemlösung: dimensionale analyse

Jede mathematische Gleichung, die separate und unterschiedliche physikalische Größen verbindet, muss dimensionskonstant sein, was bedeutet, dass sie sich an zwei Regeln halten muss. Aus diesem Grund ist das Konzept der Dimension von entscheidender Bedeutung. Die erste Regel besagt, dass die Ausdrücke einer Gleichung auf beiden Seiten einer Gleichheit die exakt gleiche Dimension haben müssen, d.h., Größen mit der gleichen Dimension können addiert oder entfernt werden. Die zweite Regel besagt, dass alle gängigen mathematischen Funktionen wie exponentielle, logarithmische und trigonometrische Funktionen dimensionlose Argumente in einer Gleichung haben müssen.

Es ist dimensionsinkonsistent, wenn eine Gleichung eine dieser beiden Regeln bricht, daher kann eine Gleichung keine genaue Aussage eines physikalischen Gesetzes darstellen. Die dimensionale Analyse kann helfen, sich an die verschiedenen Gesetze der Physik zu erinnern, algebraische Fehler oder Tippfehler zu überprüfen und sogar über die Form zu spekulieren, die zukünftige physikalische Gesetze annehmen könnten.

Die Basisgrößen können verwendet werden, um jede gewünschte physikalische Größe zu erstellen. Eine Größe wird als Produkt verschiedener Potenzen der Basisgrößen angegeben, wenn sie in Bezug auf die Basisgrößen ausgedrückt wird. Die Dimension der Größe in dieser Basis ist der Exponent der Basisgröße, die in der Gleichung erscheint.

Betrachten Sie die physikalische Größe Kraft, die definiert ist als Masse multipliziert mit Beschleunigung. Beschleunigung wird berechnet als Änderung der Geschwindigkeit geteilt durch ein Zeitintervall, während die Länge geteilt durch das Zeitintervall die Geschwindigkeit ergibt. Als Ergebnis hat die Kraft die folgenden Dimensionen: eine in Masse, eine in Länge und minus zwei in Zeit.

Problemlösung: Dimensionale Analyse

Jede mathematische Gleichung, die getrennte unterschiedliche physikalische Größen verbindet, muss dimensional konsistent sein, was bedeutet, dass sie sich an zwei halten muss ...

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Education

Lehre

JoVE Core

Physics

Physik

Unit 1

Einheit 1

Units, Dimensions, and Measurements

Einheiten, Dimensionen und Messungen

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

Problem Solving: Dimensional Analysis

dimensional checks for physics equations

Dimensional analysis helps check whether physics equations make sense. It compares the dimensions on each side of an equation and shows whether the units are consistent. This method is useful for spotting algebra errors, typos, and for remembering physical laws.

A valid equation must follow two rules. First, both sides of an equality must have the same dimension, so only quantities with matching dimensions can be added or subtracted. Second, common math functions such as exponential, logarithmic, and trigonometric functions must use dimensionless arguments.

If an equation breaks either rule, it is dimensionally inconsistent. That means the equation cannot correctly represent a physical law. Dimensional analysis also helps scientists think about the form future laws of physics might take.

Base quantities are the building blocks for other physical quantities. A quantity can be written as a product of powers of the base quantities. In that form, the dimension of a quantity is given by the exponent of each base quantity.

Force is a useful example. Force is defined as mass multiplied by acceleration. Acceleration is the change in velocity divided by time, and velocity is length divided by time. From these relationships, force has dimensions of mass to the first power, length to the first power, and time to the minus second power.

Dimensional Checks for Physics Equations

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Dimensional Checks in Physics Equations

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problem solving: dimensional analysis

Dimensional analysis helps check whether physics equations make sense. It compares the dimensions on each side of an equation and shows whether the units are consistent. This method is useful for spotting algebra errors, typos, and for remembering physical laws. A valid equation must follow two rules. First, both sides of an equality must have the same dimension, so only quantities with matching dimensions can be added or subtracted. Second, common math functions such as exponential,...

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key terms and concepts

Learn how dimensional analysis checks physics equations, keeps units consistent, and reveals the dimensions of force from mass, length, and time.

Dimensional Checks for Physics Equations | JoVE K12

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Dimensional Checks for Physics Equations

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