Question

the energy E of particle depends upon the mass m, acceleration due to gravity g and height h using dimensional analysis establish relationship between them

Answer 1

relation between E , m, g and h is given as E = mgh

you mean, we have to establish the relationship between them, right ?

we know as well,

• dimensional formula of E = [ML²T-²]
• dimensional formula of g = [LT-²]
• dimensional formula of h = [L]
• dimensional formula of m = [M]

from dimensional analysis,

let, E = m^x g^y h^z

here, x , y and z are constant terms

⇒[ML²T-²] = [M]^x[LT-²]^x [L]^z

⇒[ML²T-²] = [M^xL^(x + z) T^-2z]

on comparing both sides,

x = 1, x + z = 2 , -2 = -2z

so, x = 1 , y = 1 and z = 1

hence, E = mgh

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Answer 2

$\huge\bold\purple{Answer:-}$

you mean, we have to establish the relationship between them, right ?

we know as well,

dimensional formula of E = [ML²T-²]

dimensional formula of g = [LT-²]

dimensional formula of h = [L]

dimensional formula of m = [M]

from dimensional analysis,

let, E = m^x g^y h^z

here, x , y and z are constant terms

⇒[ML²T-²] = [M]^x[LT-²]^x [L]^z

⇒[ML²T-²] = [M^xL^(x + z) T^-2z]

on comparing both sides,

x = 1, x + z = 2 , -2 = -2z

so, x = 1 , y = 1 and z = 1

hence, E = mgh