A force is represented by \( F = ax^2 + b t^{1/2} \) Where \( x = \) distance and \( t = \) time. The dimensions of \( \frac{b^2}{a} \) are:
- \( [ML^3T^{-3}] \)
- \( [MLT^{-2}] \)
- \( [ML^{-1}T^{-1}] \)
- \( [ML^2T^3] \)
The Correct Option is A
Approach Solution - 1
To find the dimensions of \( \frac{b^2}{a} \), we need to understand the dimensional analysis of the force equation given:
The force equation is: \(F = ax^2 + b t^{1/2}\), where \( F \) is the force, \( x \) is distance, and \( t \) is time.
The dimension of force \([F]\) in terms of mass (M), length (L), and time (T) is known to be \([MLT^{-2}]\).
The dimensional formula for distance \( x \) is \([L]\), and for time \( t \) is \([T]\).
Since both terms on the right-hand side of the equation must have the same dimensions as \( F \), we can equate the dimensions:
1. For the term \(ax^2\):
\([a][L^2] = [MLT^{-2}]\).
Thus, the dimension of \( a \) is:
\([a] = [MLT^{-2}][L^{-2}] = [ML^{-1}T^{-2}]\).
2. For the term \(b t^{1/2}\):
\([b][T^{1/2}] = [MLT^{-2}]\).
Thus, the dimension of \( b \) is:
\([b] = [MLT^{-2}][T^{-1/2}] = [MLT^{-3/2}]\).
Now, we need to find the dimensions of \( \frac{b^2}{a} \):
\(\frac{b^2}{a} = \frac{[b]^2}{[a]}\)
Substitute the dimensions of \( b \) and \( a \):
\(\frac{b^2}{a} = \frac{[MLT^{-3/2}]^2}{[ML^{-1}T^{-2}]}\)
Calculate dimensions in the numerator:
\([b]^2 = [M^2L^2T^{-3}]\)
Plug into the fraction:
\(\frac{[M^2L^2T^{-3}]}{[ML^{-1}T^{-2}]} = [M^{2-1}L^{2+1}T^{-3+2}] = [ML^3T^{-1}]\)
Thus, the dimensions of \( \frac{b^2}{a} \) are \([ML^3T^{-3}]\).
The correct answer is, therefore, \([ML^3T^{-3}]\).
Approach Solution -2
Given:
\[ F = ax^2 + bt^{1/2} \]
The dimensions of \( a \) are given by:
\[ [a] = \left[ \frac{F}{x^2} \right] = [MLT^{-2}][L]^{-2} = [ML^{-1}T^{-2}] \]
The dimensions of \( b \) are given by:
\[ [b] = \left[ \frac{F}{t^{1/2}} \right] = [MLT^{-2}][T^{-1/2}] = [MLT^{-5/2}] \]
Now, the dimensions of \( \frac{b^2}{a} \) are:
\[ \left[ \frac{b^2}{a} \right] = \frac{[M^2L^2T^{-5}]}{[ML^{-1}T^{-2}]} = [ML^3T^{-3}] \]
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