Dimensions and Dimensional analysis question

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Discussion Overview

The discussion revolves around the role of dimensionless numerical factors, specifically the value of 1/2 in the kinetic energy formula E = 1/2 mv², within the context of dimensional analysis in physics. Participants explore the implications of dimensional analysis and its limitations in capturing numerical coefficients.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that the value of 1/2 does affect the formula, questioning the need for further explanation.
  • Others highlight that dimensional analysis cannot account for numerical factors like 1/2, suggesting that more advanced methods are necessary for such calculations.
  • One participant explains that dimensional analysis typically uses basic dimensions such as length, mass, and time (LMT), and notes that the dimensional formula for energy does not differentiate between kinetic and potential energy.
  • It is mentioned that dimensional formulas do not distinguish between energy density and pressure, despite both being represented by the same dimensional formula.
  • There is a discussion about the usefulness of dimensional analysis in showing relationships between different physical formulas, while noting that formulas with the same dimensional formula are not necessarily equivalent.

Areas of Agreement / Disagreement

Participants express differing views on the significance of the numerical factor 1/2 in the kinetic energy formula and the limitations of dimensional analysis, indicating that multiple competing views remain without a consensus.

Contextual Notes

Limitations of dimensional analysis are noted, particularly its inability to account for numerical coefficients and the potential for confusion between dimensionally equivalent formulas.

Zerocool97
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In this formula, E = 1/2 mv²

1/2 is not considered as it does not have any dimension. But then does it not affect the formula. Why? Explain! Please
 
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Welcome to PF;
Of course the value of 1/2 affects the formula... what is there to explain?
 
That's a limitation of dimensional analysis. numerical factors such as 1/2 in the formula for the kinetic energy cannot be calculated by dimensional analysis. Some more advanced calculation method must be used.
 
Zerocool97 said:
In this formula, E = 1/2 mv²

1/2 is not considered as it does not have any dimension. But then does it not affect the formula. Why? Explain! Please

Dimensional analysis is generally done with a few chosen "basic" dimensions. For most problems in Newtonian mechanics, length, mass and time (LMT) are sufficient. For energy, dimensions are: E=ML^{2}T^{-2}. The dimensional formula does not distinguish between kinetic and potential energy. Dimensionless values such as 1/2 in the KE formula do not appear in the dimensional formula.

Also, dimensional formulas do not distinguish between energy density and pressure. Both are represented by the formula ML^{-1}T^{-2}. The difference is that energy density is a scalar quantity and pressure is a vector quantity.

DA is useful in showing relationships between different formulas used in physics. That doesn't mean that different formulas with the same dimensional formula are necessarily strictly equivalent. They may only be dimensionally equivalent.
 
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