Question

The correctness of equation can be checked using the principle of homogeneity in dimensions.
(a) State the priciple of homogeneity.
(b) Using this principle,check whether the equation f = 2π√l/g is dimensionally correct.where f- frequency.l- length and g- acceleration due to gravity .
(c) The velocity V of a particle depends on time 't' as V = At^2+Bt . Find the dimensions and units of A and B​

Answer 1

a ) According to the principle of Homogeneity all the terms in an equation  have the same dimensional formulas .

b ) Equation ,

   $\sf F = 2\pi \sqrt{\dfrac{l}{g}}$

  Here ,

 F = Frequency

 l = Length

 g = Gravity

 $\sf \longrightarrow [M^0L^0T^-1]=\sqrt{\dfrac{[M^0L^1T^0]}{[M^0L^1T^{-2}]}} \\\\\longrightarrow [M^0L^0T^{-1}]=\sqrt{[M^0L^0T^{2}]}\\\\\longrightarrow [M^0L^0T^{-1}]=[M^0L^0T^{2}]^{\frac{1}{2}}\\\\\longrightarrow [M^0L^0T^{-1}] \neq [M^0L^0T^{1}]$

 The equation is not dimensionally correct .

c ) V = At²+Bt

    Here ,

    V = Velocity

    t = Time

    According to the principle of Homogeneity all the terms in an equation have the same dimensional formulas , So

    ⇒ V = At²

      $\sf\longrightarrow A=\dfrac{V}{t^2}$

                $\sf = \dfrac{[M^0L^1T^{-1}]}{[M^0L^0T^1]^2}$

                $\sf = \dfrac{[M^0L^1T^{-1}]}{[M^0L^0T^2]}$

                $\sf\bf = [M^0L^1T^{-3}]$

     ⇒ V = Bt

       $\sf \longrightarrow B= \dfrac{V}{t}$

                 $\sf = \dfrac{[M^0L^1T^{-1}]}{[M^0L^0T^1]}$

                 $= \bf [M^0L^1T^{-2}]$

HOPE IT HELPS U .......  :)

Answer 2

Answer:

a) the magnitude of physical quantities may be added together or subtract from one another only if they have the same dimensions

b) this is dimensionally correct

c)