Question

here is a question in physics
find the correctness of the equation using Diamentional analysis
1/2 mv2= hv+mgh2​

Answer 1

Correct Question:

Find the correctness of the equation using Diamentional analysis.

1/2 mv2= hv + mgh

Solution:

We have to find the correctness of the equation using Dimensional analysis.

Equation: 1/2 mv² = h'ν' + mgh

Frequency is represented by v' instead of v and Planck’s constant by h'.

Frequency (v') = 1/(Time period)

= 1/[T¹]

= [T-¹]

Planck’s constant (E) = h'v'

= [M¹L²T-²] / [T-¹]

= [M¹L²T-¹]

Here;

m = mass, v = velocity, h = height and g = acceleration due to gravity and ν' = frequency

Also, 1/2 is constant. So, neglected

We left with mv² = h'v' + mgh

Now,

[M¹] [L¹T-¹]² = [M¹L²T-¹] [T-¹] + [M¹] [L¹T-²] [L¹]

[M¹] [L²T-²] = [M¹L²T-²] + [M¹] [L²T-²]

[M¹L²T-²] = [M¹L²T-²] + [M¹L²T-²]

L.H.S. = R.H.S.

Hence, the equation is correct.

Answer 2

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CORRECT QUESTION :

here is a question in physics.

find the correctness of the equation using Diamentional analysis .

1/2 mv2= hv+mgh.

SOLUTION :

In the above Question, we have to see if the above equation is Dimensionally correct or not.

The given Equation is :

$\dfrac{1}{2} m { v } ^ 2 = h { \nu } + mgh$

Let's see the LHS first.

( 1 / 2 ) is dimension less.

$V = L . { T } ^ { -1 }$

$LHS = M .{ [ L . { T } ^ { -1 } ] } ^ 2 = M . { L } ^ 2 . { T } ^ { -2 }$

RHS -

Using the Equation of Homogeneity -

Dimension of hv should be equal to the dimension of mgH which should be equal to the dimension of the RHS.

Let us find the dimension of $H { \nu }$

Here,

H is the plank " s constant.

${ \nu }$ is the frequency.

$H = M . { L } ^ { 2 } . { T } ^ { -1 }$

${ \nu } = { T } ^ { -1 }$

$\therefore { H . { \nu } = M . { L } ^ { 2 } . { T } ^ { -2 } }$

Hence, the dimension of the LHS is equal to the dimension of The RHS.

Hence the above Equation is Dimensionally Correct.