1.12
All physical quantities can be expressed using either base quantities or derived quantities and each quantity is represented by a symbol, which defines its dimensions.
For instance, the speed of a car is defined as the distance divided by time. The term distance corresponds to the quantity length, denoted with L and time with T.
Hence, we can write the dimension of the quantity speed, as L divided by T or LT to the power of minus one.
For an equation to be dimensionally correct, it should obey two rules. Number one, the expressions on each side of the equality in an equation must have the same dimensions.
Number two, the standard mathematical functions in equations must be dimensionless
For example, we know the dimension of volume is L cubed. Now, consider a cylinder with radius r and height h.
We know that the volume of a cylinder is π r squared h. The term π is a constant, and it's a dimensionless quantity. The term r corresponds to the quantity length, and we can write its dimension as L squared, and the term h also corresponds to the quantity length, which gives the dimension of the volume of the cylinder as L cubed. Hence, the equation is dimensionally correct.
As long as we know the dimensions of the individual physical quantities that appear in an equation, we can check to see whether the equation is dimensionally consistent.
Another application of dimensional analysis is to remember an equation. For example, let's say that you don't remember whether speed equals time divided by distance or distance divided by time.
The dimensions of time, distance, and speed are T, L, and LT to the power of minus one respectively. Reducing both the equations to their fundamental units on each side of the equation, we get speed equals distance divided by time.
Het concept van dimensie is belangrijk omdat elke wiskundige vergelijking die fysische grootheden koppelt, dimensioneel consistent moet zijn, wat impliceert dat wiskundige vergelijkingen aan de volgende twee regels moeten voldoen. De eerste regel is dat in een vergelijking de uitdrukkingen aan elke zijde van het gelijkteken dezelfde dimensies moeten hebben. Dit is vrij intuïtief, aangezien we alleen grootheden van hetzelfde type (dimensie) kunnen optellen of aftrekken. De tweede regel stelt dat in een vergelijking de argumenten van een van de standaard wiskundige functies zoals goniometrische functies, logaritmen of exponentiële functies dimensieloos moeten zijn.
Als een van deze twee regels wordt geschonden, is de vergelijking dimensioneel inconsistent en kan ze geen correcte weergave van een natuurkundige wet zijn. Dimensionale analyse kan worden gebruikt om fouten of typefouten in algebra op te sporen, om de verschillende natuurwetten beter te onthouden en zelfs om de vorm die nieuwe natuurwetten kunnen aannemen te suggereren.
Laten we het effect van de operaties van de calculus op dimensies begrijpen. De afgeleide van een functie is de helling van de raaklijn aan de grafiek, en hellingen zijn verhoudingen. Dus voor fysieke grootheden, bijvoorbeeld v en t, is dit de dimensie van de afgeleide van v met betrekking tot t is de verhouding tussen de dimensie van v en die van t. Op dezelfde manier, aangezien integralen slechts sommen van producten zijn, is de dimensie van de integraal van v met betrekking tot t eenvoudigweg de dimensie van v maal de dimensie van t.
Deze tekst is een bewerking van Openstax, University Physics Deel 1, Sectie 1.4: Dimensionale Analyse.
All physical quantities can be expressed using either base quantities or derived quantities and each quantity is represented by a symbol, which defines its dimensions.
For instance, the speed of a car is defined as the distance divided by time. The term distance corresponds to the quantity length, denoted with L and time with T.
Hence, we can write the dimension of the quantity speed, as L divided by T or LT to the power of minus one.
For an equation to be dimensionally correct, it should obey two rules. Number one, the expressions on each side of the equality in an equation must have the same dimensions.
Number two, the standard mathematical functions in equations must be dimensionless
For example, we know the dimension of volume is L cubed. Now, consider a cylinder with radius r and height h.
We know that the volume of a cylinder is π r squared h. The term π is a constant, and it's a dimensionless quantity. The term r corresponds to the quantity length, and we can write its dimension as L squared, and the term h also corresponds to the quantity length, which gives the dimension of the volume of the cylinder as L cubed. Hence, the equation is dimensionally correct.
As long as we know the dimensions of the individual physical quantities that appear in an equation, we can check to see whether the equation is dimensionally consistent.
Another application of dimensional analysis is to remember an equation. For example, let's say that you don't remember whether speed equals time divided by distance or distance divided by time.
The dimensions of time, distance, and speed are T, L, and LT to the power of minus one respectively. Reducing both the equations to their fundamental units on each side of the equation, we get speed equals distance divided by time.
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