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Units and Measurements — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsUnits and Measurements1 Mark
Question
The orbital velocity v of a satellite may depend on its mass m, distance r from the centre of earth and acceleration due to gravity g. Obtain an expression for orbital velocity.
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Answer

Suppose orbital velocity of satellite be given by the relation: $\text{v}=\text{km}^{\text{a}}\text{r}^{\text{b}}\text{g}^\text{c}$ E v =kmorogo where, k is a dimensionless constant and a, b, c are unknown powers. Writing dimensions on two sides of equation, We have: $[\text{M}^{0}\text{L}^{1}\text{T}^{-1}]=[\text{M}]^{\text{a}}[\text{L}]^{\text{b}}[\text{LT}^{-2}]^{\text{C}}=[\text{M}^{\text{a}}\text{L}^{\text{b}+\text{c}}\text{T}^{-2\text{c}}]$ Applying principle of homogeneity of dimensional equation, We find that: $\text{a}=0\Rightarrow\text{b}+\text{c}=1\Rightarrow-2\text{c}=-1$ On solving these equations We find that: $\text{a}=0,\text{b}=+\frac{1}{2}\text{ and }\text{c}=+\frac{1}{2}$ $\text{v}=\text{kr}^{\frac{1}{2}}\text{g}^{\frac{1}{2}}\Rightarrow\text{v}=\text{k}\sqrt{\text{rg}}$

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