B Must two quantities have the same dimensions

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Two quantities must have the same dimensions when one is used as an exponent for the other, as the exponent must be dimensionless. In the expression mv²/r, the term v² indicates that velocity is multiplied by itself, reinforcing that the exponent is dimensionless. Arguments for trigonometric functions and logarithms must also be dimensionless. Understanding dimensions can be challenging, especially with concepts like acceleration, which can be expressed in different forms. Ultimately, clarity in dimensional analysis is crucial for accurate physical interpretations.
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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