In a system of units, let the units of force energy and velocity be 10 N,0.1 J and 100 m/s respectively. In this new system find the unit of length,mass and time
 

Asked by Anil | 21st May, 2017, 09:27: PM

Expert Answer:

begin mathsize 12px style Let space straight M subscript 1 space comma space straight L subscript 1 space space and space straight T subscript 1 space space be space the space units space of space mass comma space length space and space time space in space SI space and
straight M subscript 2 space comma straight L subscript 2 space space and space straight T subscript 2 space space the space corresponding space units space in space new space system.
Dimensional space formula space for space force space is space open square brackets straight M to the power of 1 space straight L to the power of 1 space straight T to the power of – 2 end exponent close square brackets
rightwards arrow Conversion space formula space for space force space becomes
straight n subscript 2 space space equals space straight n subscript 1 space open square brackets straight M subscript 1 over straight M subscript 2 close square brackets to the power of 1 space open square brackets straight L subscript 1 over straight L subscript 2 close square brackets to the power of 1 space open square brackets straight T subscript 1 over straight T subscript 2 close square brackets to the power of negative 2 end exponent
Given space that space here space straight n subscript 1 space equals space 10 space straight N space and space straight n subscript 2 space equals space 1
therefore space 1 space space equals space 10 space open square brackets straight M subscript 1 over straight M subscript 2 close square brackets to the power of 1 space open square brackets straight L subscript 1 over straight L subscript 2 close square brackets to the power of 1 space open square brackets straight T subscript 1 over straight T subscript 2 close square brackets to the power of negative 2 end exponent space space......... space left parenthesis Equation space 1 right parenthesis
Dimensional space formula space for space work space is space open square brackets straight M to the power of 1 space straight L squared space straight T to the power of – 2 end exponent close square brackets
rightwards arrow Conversion space formula space for space energy space becomes
straight n subscript 2 space space equals space straight n subscript 1 space open square brackets straight M subscript 1 over straight M subscript 2 close square brackets to the power of 1 space open square brackets straight L subscript 1 over straight L subscript 2 close square brackets squared space open square brackets straight T subscript 1 over straight T subscript 2 close square brackets to the power of negative 2 end exponent
Given space that space here space straight n subscript 1 space equals space 5 space straight J space and space straight n subscript 2 space equals space 1
1 space space equals space 5 space open square brackets straight M subscript 1 over straight M subscript 2 close square brackets to the power of 1 space open square brackets straight L subscript 1 over straight L subscript 2 close square brackets squared space open square brackets straight T subscript 1 over straight T subscript 2 close square brackets to the power of negative 2 end exponent space space......... space left parenthesis Equation space 2 right parenthesis
Dimensional space formula space for space velocity space is space open square brackets straight M to the power of 0 space straight L to the power of 1 space straight T to the power of – 1 end exponent close square brackets
rightwards arrow Conversion space formula space for space velocity space becomes
straight n subscript 2 space space equals space straight n subscript 1 space open square brackets straight M subscript 1 over straight M subscript 2 close square brackets to the power of 0 space open square brackets straight L subscript 1 over straight L subscript 2 close square brackets to the power of 1 space open square brackets straight T subscript 1 over straight T subscript 2 close square brackets to the power of negative 1 end exponent
Given space that space here space straight n subscript 1 space equals space 0.5 space straight J space and space straight n subscript 2 space equals space 1
1 space space equals space 0.5 space open square brackets straight L subscript 1 over straight L subscript 2 close square brackets to the power of 1 space open square brackets straight T subscript 1 over straight T subscript 2 close square brackets to the power of negative 1 end exponent space......... space left parenthesis Equation space 3 right parenthesis
Dividing space equation space left parenthesis 2 right parenthesis space by space space equation space left parenthesis 1 right parenthesis comma space we space get
straight L subscript 2 space equals space straight L subscript 1 over 2 space equals space 1 half space straight m space equals space 0.5 space straight m
therefore straight L subscript 1 over straight L subscript 2 space equals space 2
Substituting space this space value space in space equation space left parenthesis 3 right parenthesis comma space we space get
1 space space equals space 0.5 space cross times space 2 space cross times space open square brackets straight T subscript 1 over straight T subscript 2 close square brackets to the power of negative 1 end exponent
rightwards arrow straight T subscript 1 over straight T subscript 2 space equals space 1
therefore space straight T subscript 2 space equals space 1 straight s
Substituting space this space value space of space open square brackets straight L subscript 1 over straight L subscript 2 close square brackets space and space open square brackets straight T subscript 1 over straight T subscript 2 close square brackets space in space equation space left parenthesis 1 right parenthesis comma space we space get
1 space space equals space 10 space open square brackets straight M subscript 1 over straight M subscript 2 close square brackets to the power of 1 space cross times space 2 space cross times space 1 space
rightwards arrow space 1 space equals space 20 open square brackets straight M subscript 1 over straight M subscript 2 close square brackets
rightwards arrow straight M subscript 2 space equals space 20 space straight M subscript 1
As space straight M subscript 1 space equals space 1 space kg
therefore space straight M subscript 2 space equals space 20 space kg space
Hence space units space of space mass comma space length space and space time space are space 20 space kg comma space 0.5 space straight m space and space 1 space sec space respectively.


end style

Answered by Yashvanti Jain | 22nd May, 2017, 12:00: PM