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A physical quantity force F and density rho are related by the expression F = alpha / (beta + sqrt(rho)). If the dimension of length in alpha is 'x' and the dimension of length in beta is 'y', find the value of |x + y|.

1. 1
2. 2
3. 3
4. 4

Correct answer: 4

Solution

Density rho has dimensions M L⁻³. So sqrt(rho) has dimensions M^(1/2) L^(-3/2). Beta must match: [beta] = M^(1/2) L^(-3/2), so length power in beta is y = -3/2. Alpha = F * beta = [M L T⁻²] * [M^(1/2) L^(-3/2)] = M^(3/2) L^(-1/2) T⁻², so x = -1/2. |x+y| = |-1/2 + (-3/2)| = |-2| = 2. Hmm, that gives 2. Alternatively if the formula is F = alpha/(beta + rho): [beta]=[rho]=ML⁻³ => y=-3; [alpha]=[F*rho]=MLT⁻²*ML⁻³=M²L⁻²T⁻² => x=-2; |x+y|=5. Let me try F=alpha/(beta+rho^(1/2)) giving the sqrt interpretation: answer 2.

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