Question

Energy = ½ MV2 . How can you help him with the help of dimensional analysis

Answer 1

An equation has been provided to us relating energy , mass and velocity. we need to find out if the equation is dimensionally correct or not.

For this , we shall trying to find out the dimensions of LHS and RHS and see if they are same.

For dimensions we shall try to express each of the terms into the fundamental quantities like mass ,length ,time , etc.

LHS :

Energy

$= work$

$= force \times displacement$

$= mass \times acc. \times displacment$

$= \{M \} \times \{L {T}^{ - 2} \} \times \{L \}$

$= \{M {L}^{2} {T}^{ - 2} \}$

RHS :

½ MV²

$= mass \times {(velocity)}^{2}$

$= \{M \} \times { \{L {T}^{ - 1} \}}^{2}$

$= \{M \} \times \{ {L}^{2} {T}^{ - 2} \}$

$= \{M {L}^{2} {T}^{ - 2} \}$

So , LHS and RHS are same and hence equation is dimensionally correct.

Answer 2

Given :

Energy = ½ mv²

To Find :

Dimensional analysis of this equation.

Concept :

To prove a dimensional equation, we need the R.H.S and L.H.S of the dimensional equation to be equal.

Solution :

R.H.S ( energy) :

Force ×displacement

Mass ×acceleration ×displacement

M× LT¯²× L

ML²T¯² ________________EQUATION 1

And, L.H.S (½mv²):

M × velocity ²

M ×(LT¯¹)²

ML²T¯²_______________EQUATION 2

Hence, R.H.S = L.H.S. So the given equation Energy =½mv² is dimensionally correct.