Answer:
1. Given Expression: (f = \frac{1}{2}l\sqrt{\frac{f \cdot l}{m}})
2. Identify the Variables:
- (f) represents some physical quantity.
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.
