Dimensional analysis for spring-mass frequency

Steps:

• Recall the standard formula for vibration frequency: $F = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$.
• Rewrite as $F \propto m^{-1/2} k^{1/2}$, since the constant $\frac{1}{2\pi}$ is dimensionless.
• Compare to $F = C m^p k^q$, so $p = -1/2$ and $q = 1/2$.
• Verify dimensions: [F] = T^{-1}, [m] = M, [k] = M T^{-2}; thus T^{-1} = M^{p+q} T^{-2q}, yielding p + q = 0 and -2q = -1.

Why B is correct:

• Matches the formula $F \propto \sqrt{k/m}$, where frequency inversely scales with mass to the power of 1/2 and directly with spring constant to the power of 1/2.

Why the others are wrong:

• A: Incorrect exponents; would imply frequency decreases with both mass and stiffness, violating Hooke's law.
• C: Swapped exponents; describes period, not frequency.
• D: Both positive; implies frequency increases with mass, contradicting inverse square root dependence.

Final answer: B