Question

state the principle of homogeneity of dimensions. Test the dimensional homogeneity of the following equation: h=h0+ v0t +1/2 gt^2.

Answer 1

$<b><i>$
$Principle:$
The principle of homogeneity of dimensions states that if a relation is true then dimensions of each term on both the side of equation are the same.

$Testing\: the\: dimensional\: Homogeneity\: for\:$ $h =h_{0} + v_{0}t+\frac{1}{2}gt^{2}$:

We have,
As per dimensional Notation,

$h = L$
$v = LT^{-1}$
$t = T$
$g = LT^{-2}$

LHS =
$L$

RHS=
$L + [LT^{-1}×T] + \frac{1}{2}[LT^{-2} × T^{2}]$

= $L + L + \frac{1}{2}[L]$

= $L$
Since All are of Dimension L , the result of sum is also equals L

Therefore,

$LHS\: =\: RHS$ Dimensionally

So the Relation is correct.

✌✌✌

Answer 2

Answer:

The principle of homogeneity states that the dimensions of each the terms of a dimensional equation on both sides are the same.

Using this principle the given equation will have same dimension on both sides.

On left side : h=[L], dimesion of legth

On right side: h

0

=[L],v

0

t=[LT

−1

][T]=[L],gT

2

=[LT

−2

][T

2

]=[L]

Thus, the dimension on both sides quantities are same.