Question

Force F and density d are related as F = ∝/β root d then find the dimensions of ∝ and β.

Answer 1

Answer:

Given;

$\longrightarrow\:\:\tt F = \dfrac{\alpha}{\beta + \sqrt{D}}$

$\dag\: \underline{\underline{\textsf{Dimensions of Force :}}}$

$\longrightarrow\:\:\sf Force = mass \times Acceleration$

$\longrightarrow\:\:\sf Force = [M][LT^{-2}]$

$\longrightarrow\:\:\sf Force = [MLT^{-2}]$

$\dag\: \underline{\underline{\textsf{Dimensions of Density :}}}$

$\longrightarrow\:\:\sf Density = \dfrac{Mass}{Volume}$

$\longrightarrow\:\:\sf Density = \dfrac{[M]}{[L^3]}$

$\longrightarrow\:\:\sf Density = [M][L^{ - 3}]$

$\longrightarrow\:\:\sf Density = [ML^{ - 3}]$

Now,

$\longrightarrow\:\:\tt [F] =[ \alpha ]$

$\longrightarrow\:\: \underline{ \boxed{\tt [ \alpha ]= [MLT^{-2}]}}$

Also,

$\longrightarrow\:\:\tt [\beta]= [\sqrt{D}]$

$\longrightarrow\:\:\tt [\beta]= [D]^{ \frac{1}{2} }$

$\longrightarrow\:\:\tt [\beta]= [ML^{ - 3}]^{ \frac{1}{2} }$

$\longrightarrow\:\: \underline{ \boxed{\tt [\beta]= [M^{ \frac{1}{2} } L^{ \frac{ - 3}{2} }] }}$