Answer:
Proof in explanataion
Explanation:
The basic dimensions are as follows:
MASS = M
LENGTH = L
TIME = T
i)
Given equation is:
[tex]H = \frac{u^2Sin^2\phi}{2g}[/tex]
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,
[tex][L] = \frac{[LT^{-1}]^2}{[LT^{-2}]}\\\\\ [L] = [L^{(2-1)}T^{(-2+2)}][/tex]
[L] = [L]
Hence, the equation is proven to be homogenous.
ii)
[tex]F = \frac{Gm_1m_2}{r^2}\\\\[/tex]
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,
[tex][MLT^{-2}] = \frac{[M^{-1}L^{3}T^{-2}][M][M]}{[L]^2}\\\\\ [MLT^{-2}] = [M^{(-1+1+1)}L^{(3-2)}T^{-2}]\\\\[/tex]
[MLT⁻²] = [MLT⁻²]
Hence, the equation is proven to be homogenous.
