omega^2 = T/mL The formula above expresses the square of the angular frequency omega of a vibrating system in terms...

1. TRANSLATE the problem requirement

• Given formula: \(\omega^2 = \frac{T}{mL}\)
• Goal: Solve for m in terms of the other variables \(\omega, T, L\)
• This means m should be isolated on one side of the equation

2. SIMPLIFY through algebraic manipulation

• Start with: \(\omega^2 = \frac{T}{mL}\)
• To clear the fraction, multiply both sides by mL:
  \(\omega^2 \times mL = T\)
• This gives us: \(\omega^2mL = T\)

3. SIMPLIFY to isolate m

• We have: \(\omega^2mL = T\)
• To get m by itself, divide both sides by \(\omega^2L\):
  \(m = \frac{T}{\omega^2L}\)

Answer: (A) \(m = \frac{T}{\omega^2L}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students get confused about which operations to perform and in what order when dealing with the fraction \(\frac{T}{mL}\). They might incorrectly try to 'cross multiply' or make errors when clearing the denominator.

For example, they might incorrectly multiply only one side by mL, or forget to divide by both \(\omega^2\) AND L, leading them to select Choice (C) \(m = \frac{TL}{\omega^2}\) or Choice (D) \(m = \frac{L}{\omega^2T}\).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what 'solve for m' means and might try to substitute values or rearrange incorrectly, potentially leading to the reciprocal relationship in Choice (B) \(m = \frac{\omega^2L}{T}\).

The Bottom Line:

This problem tests pure algebraic manipulation skills. Success requires systematic step-by-step isolation of the target variable while maintaining equation balance.