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Grade -11 (NEB) > Science Physics Dimensional Analysis and Applications, Physical Quantities

Is this escape velocity expression, dimensionally correct?

A student writes expression for escape velocity of earth, \(v=\sqrt{\frac R{2GM}}\). Check the correctness of the formula by using dimensional analysis.

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Solution:

Here, given equation is, \(v=\sqrt{\frac R{2GM}}\)

Dimension of LHS

= \(\left[LT^{-1}\right]\)

Dimension of RHS

= \(\left(\frac{\left[L\right]}{2\left[M\right]\left[M^1L^3T^2\right]}\right)^\frac12\)

= \(\frac1{\sqrt2}\left[L^{-2}T^2\right]^\frac12\)

= \(\frac1{\sqrt2}\left[L^{-1}T\right]\)

= \(\left[L^{-1}T\right]\)

Here, \(\frac1{\sqrt2}\) is dimensionless constant.

So, Dimension of LHS ≠ Dimension of RHS

Thus, Given equation is incorrect.

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Things to Remember From Physical Quantities

  • The four applications of dimensional analysis are:
    • To check the correctness of physical relation.
    • To derive the relation between various physical quantities.
    • To convert the value of physical quantities from one system of units into another system of units.
    • To find the dimensions of constants in the given equation.

  • The correctness of physical relation can be determined by comparing the dimension of the left-hand side (LHS) and the dimension of the right-hand side (RHS)
  • We can use the formula, \(N_2=N_1\left[\frac{M_1}{M_2}\right]^a\left[\frac{L_1}{L_2}\right]^b\left[\frac{T_1}{T_2}\right]^c\)\) to convert unit from one to another system using dimension.

  • Despite the usefulness of dimensions, there are some limitations. They are:
    • The dimensional analysis does not give any information about dimensionless constants.
    • If the quantity depends on more than three other physical quantities having dimensions, the formula cannot be derived.
    • We cannot derive the formula containing trigonometric functions, logarithmic functions, exponential functions, etc. It is best suited for linear functions only.
    • The exact form of a relationship cannot be determined when there is more than one part in any relationship.
    • It gives no information about the physical quantity,

 

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