High School Must two quantities have the same dimensions

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SUMMARY

In physics, when raising a quantity to a power, the exponent must be dimensionless. This principle is crucial for maintaining dimensional consistency in equations. For example, in the expression mv²/r, the term v² represents velocity squared, which must be interpreted correctly to ensure that the dimensions align. The discussion emphasizes that quantities like velocity, when squared, can be understood in terms of their dimensional equivalents, such as length squared representing area.

PREREQUISITES
  • Understanding of dimensional analysis
  • Familiarity with physical quantities and their dimensions
  • Basic knowledge of kinematics, particularly velocity and acceleration
  • Concept of dimensionless quantities in mathematical expressions
NEXT STEPS
  • Study dimensional analysis techniques in physics
  • Explore the concept of dimensionless numbers in fluid dynamics
  • Learn about the implications of dimensional consistency in physical equations
  • Investigate the relationship between velocity, acceleration, and their dimensional representations
USEFUL FOR

Students of physics, educators teaching dimensional analysis, and professionals in engineering fields who require a solid understanding of dimensional consistency in mathematical modeling.

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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