Question

Derive dimensional formula of time with length, mass and gravity

Answer 1

Q : Express the dimension of time in terms of Dimension of Length, Mass and Gravity.
Solution. :
Steps and Understanding :
1)
Let the
dim(mass ,m) = M
dim(length ,l) = L
dim(gravity,g ) = L/T^2
dim(Time ,t) = T

2)
Then,
$t = k \times {m}^{x} \times {l}^{y} \times {g}^{z}$


where k is some constant.

3)
$dim(t) = dim(k) \times dim( {m}^{x} ) \\ \times dim( {l}^{y} ) \times dim( {g}^{z} )$

Since, constant k is dimensionless.

4) T = (M^x)*(L^y) *( L/T^2)^z
=> 2z = -1 & y+z= 0 , x =0
$= > z = \frac{ - 1}{2} \\ = > y = \frac{1}{2} \\ = > x = 0$

5) Now,
G= dim(acceleration due to gravity) = dim (g)

T =( L/G )^(1/2)


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