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Answer:

1.  Given Expression: (f = \frac{1}{2}l\sqrt{\frac{f \cdot l}{m}})

2.  Identify the Variables:

  • (f) represents some physical quantity.
  • (l) is the length.
  • (m) is the mass.

3.  Write the Dimensions:

  • Dimension of length ((l)): ([L])
  • Dimension of mass ((m)): ([M])
  • Dimension of frequency ((f)): ([T^{-1}]) (since frequency is the reciprocal of time)

4.  Analyze the Given Expression:

The left-hand side (LHS) of the equation is (f), which has the dimension ([T^{-1}]).

The right-hand side (RHS) involves several terms:

  • (\frac{1}{2}l) has the same dimension as (l), which is ([L]).
  • (\sqrt{\frac{f \cdot l}{m}}) has the dimension of (\sqrt{\frac{T^{-1} \cdot L}{M}}), which simplifies to ([T^{-1}]).

5.  Dimensional Consistency:

Since the LHS and RHS have the same dimension ([T^{-1}]), the formula is dimensionally consistent.

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