Question

By method of dimension show that the following equation are homogenous.

Answer 1

Answer:

Proof in explanataion

Explanation:

The basic dimensions are as follows:

MASS = M

LENGTH = L

TIME = T

i)

Given equation is:

$H = \frac{u^2Sin^2\phi}{2g}$

where,

H = height (meters)

u = speed (m/s)

g = acceleration due to gravity (m/s²)

Sin Ф = constant (no unit)

So there dimensions will be:

H = [L]

u = [LT⁻¹]

g = [LT⁻²]

Sin Ф = no dimension

Therefore,

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

[L] = [L]

Hence, the equation is proven to be homogenous.

ii)

$F = \frac{Gm_1m_2}{r^2}$

where,

F = Force = Newton = kg.m/s² = [MLT⁻²]

G = Gravitational Constant = N.m²/kg² = (kg.m/s²)m²/kg² = m³/kg.s²

G = [M⁻¹L³T⁻²]

m₁ = m₂ = mass = kg = [M]

r = distance = m = [L]

Therefore,

$[MLT^{-2}] = \frac{[M^{-1}L^{3}T^{-2}][M][M]}{[L]^2}\\\\\ [MLT^{-2}] = [M^{(-1+1+1)}L^{(3-2)}T^{-2}]$

[MLT⁻²] = [MLT⁻²]

Hence, the equation is proven to be homogenous.