Question

Energy of an object depends on mass height and gravirational accekeration find fomula using dimensions analysi

Answer 1

$\huge\text{\underline{Answer}}$

$\sf{\underline{Given}}$

E $\alpha$ $\bold{ {m}^{a} {h}^{b} {g}^{c} }$

where m is mass g is acc. due to gravity and h is height.

The dimensional formula is E = $\bold{m {l}^{2} {t}^{ - 2}}$

$\bold{E = [{m}^{a} {h}^{b} {g}^{c}]}$

The dimensional formula of mass is m.

The dimensional formula for h is l.

The dimensional formula for acc. due to gravity is $\bold{l {t}^{ - 2} }$

$\bold{[M{L}^{2} {T}^{-2}] ={ [M]}^{a} {[l]}^{b} {[l {t}^{ - 2} ]}{c}}$

$\bold{m {l}^{2} {t}^{ - 2} = \: {m}^{a} {l}^{b + c} {t}^{ - 2c} }$

on comparing R. H. S and L. H. S

a= 1 , 2 = b + c , -2 = -2c

after solving we get,

a = 1 , b = 1 , c = 1

Now put the value of a, b and c we get

$\bold{E = m g h }$