Must two quantities have the same dimensions

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Discussion Overview

The discussion revolves around whether two quantities must have the same dimensions when one is used as an exponent to raise another to a power. It explores the implications of dimensional analysis in mathematical expressions, particularly in the context of physics equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions if two quantities must share the same dimensions when one is an exponent of the other, specifically referencing the term '2' in the expression mv²/r.
  • Another participant asserts that the exponent must be dimensionless, clarifying that v² represents v multiplied by itself.
  • A third participant reiterates the necessity for exponents to be dimensionless and extends this to the arguments of trigonometric functions and logarithms, which also must be dimensionless.
  • A later reply reflects on the difficulty of conceptualizing dimensions for squared quantities, contrasting it with more straightforward examples like area and volume, and discusses the interpretation of acceleration as a change in velocity over time.

Areas of Agreement / Disagreement

Participants generally agree that exponents must be dimensionless, but there is no consensus on the broader implications of dimensionality in the context of the original question.

Contextual Notes

Some participants express uncertainty about the interpretation of squared quantities and their dimensions, indicating a need for further clarification on dimensional analysis in various contexts.

Nikhil faraday
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Must two quantities have the same dimensions if you are using one quantity as an exponent in raising other to a power?
What is the dimension ( or dimensionless) of '2' in mv2/r ?( v is raised to the second power)
 
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The exponent should be dimensionless. The simple answer is: v2 means v times v.
 
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Nikhil faraday said:
Must two quantities have the same dimensions if you are using one quantity as an exponent in raising other to a power?
To second what @scottdave said, the exponent must be dimensionless. Also the argument to a sin or cos or other trig function or a log must all be dimensionless.
 
I admit that it is difficult to think of what (velocity)2 could mean. (Length)2 is easy enough to understand that is Area. (length)3is volume.
Even Acceleration (meters / sec2) might at first seem baffling, but you can rewrite it as (meters per second) per second.

Think of it like this: 9.8 meters per second per second. What does it mean? Every second, a falling object gains another 9.8 m/s of velocity.
 

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