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

Check the correctness of the relation, h = 2Tcos(θ) / r ρ g, using dimension

Check the correctness of the relation, \(h=\frac{2T\cos\left(\theta\right)}{r\rho g}\), where symbols have usual meaning.

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Tribhuvan University > Institute of Science and Technology > B.Sc. (CSIT)

Solution:

Here, the given formula is \(h=\frac{2T\cos\left(\theta\right)}{r\rho g}\)

Here,

the dimension of h = [L]

the dimension of r = [L]

the dimension of ρ = [ML-3]

the dimension of g = [LT-2]

the dimension of T = [MT-2] and 2 cos(θ) is dimensionless.

Now the dimension of the quantity of the left-hand side is [L]

And, the dimension of the quantity of the right-hand side is

\(\frac{2T\cos\left(\theta\right)}{r\rho g}=\frac{\left[ML^{-2}\right]}{\left[L\right]\left[ML^{-3}\right]\left[LT^{-2}\right]}=\left[M^{1-1}L^{-1+3-1}T^{-2+2}\right]=\left[L\right]\)

The, dimension on left hand side are equal to the dimensions to the right side of the formula. So, the formula is dimensionally correct.

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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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