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

Suponha que um pêndulo simples de massa m esteja ligado a uma corda de comprimento L, oscilando sob a influência da gravidade g. Qual é a forma da equação para o período de tempo do pêndulo?
Inicialmente, identifique e liste as variáveis envolvidas no problema. O período de tempo T pode ser expresso como o produto dessas variáveis, cada uma elevada a um expoente desconhecido. Aqui, k é uma constante adimensional.
Excluindo a constante adimensional, obtém-se uma equação relacionando as dimensões das variáveis com o período de tempo.
Agora, igualando os expoentes das dimensões em ambos os lados e resolvendo as equações, os valores dos expoentes desconhecidos são determinados.
Ao substituir os expoentes, obtém-se a expressão final para o período de tempo, que é um produto da constante k e da raiz quadrada do comprimento sobre a aceleração gravitacional.
Uma das limitações da análise dimensional é que ela não nos permite encontrar o valor da constante adimensional k.

<ol>
	<li>OpenStax. (2019). Física Universitária Vol. 1. [Versão Web]. Obtido de <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>; seção 1.4; páginas 22-25.</li>
</ol>

resolução de problemas: análise dimensional

Cada equação matemática que conecta grandezas físicas distintas deve ser dimensionalmente consistente, o que implica que ela deve obedecer a duas regras. Por esta razão, o conceito de dimensão é crucial. A primeira regra é que as expressões de uma equação de cada lado de uma igualdade devem ter exatamente a mesma dimensão, ou seja, quantidades de mesma dimensão podem ser adicionadas ou removidas. A segunda regra estipula que todas as funções matemáticas populares, como funções exponenciais, logarítmicas e trigonométricas, devem ter argumentos adimensionais em uma equação.

Uma equação que quebre uma dessas duas regras é dimensionalmente inconsistente e, portanto, essa equação não pode ser uma representação precisa de qualquer lei física. A análise dimensional pode ajudar a lembrar as diferentes leis da física, verificar erros na resolução, ou erros de digitação, e até mesmo especular sobre a forma que futuras leis da física podem ter.

As grandezas de base podem ser usadas para criar quaisquer quantidades físicas desejadas. Uma grandeza é expressa como o produto de várias potências das grandezas de base quando é expressa em termos das grandezas de base. A dimensão da grandeza naquela base é o expoente de uma grandeza de base que aparece na equação.

Considere a grandeza física força, que é definida como massa multiplicada pela aceleração. A aceleração é calculada como a variação da velocidade dividida por um intervalo de tempo, enquanto o comprimento dividido pelo intervalo de tempo equivale à velocidade. Como resultado, a força tem as seguintes dimensões: uma em massa, uma em comprimento e menos duas em tempo.

Resolução de Problemas: Análise Dimensional

Toda equação matemática que conecta quantidades físicas distintas separadas deve ser dimensionalmente consistente, o que implica que deve obedecer a dois ...

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Education

JoVE Core

Physics

Física

Unit 1

Unidade 1

Units, Dimensions, and Measurements

Unidades, Dimensões e Medidas

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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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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

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Dimensional Checks for Physics Equations | JoVE K12

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Dimensional Checks for Physics Equations

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