v^2 = LT/m The formula above expresses the square of the speed v of a wave moving along a string...

1. TRANSLATE the problem requirement

• Given: \(\mathrm{v^2 = \frac{LT}{m}}\)
• Find: T in terms of m, v, and L
• What this means: Rearrange the equation so T stands alone on one side

2. SIMPLIFY through algebraic manipulation

• Current equation: \(\mathrm{v^2 = \frac{LT}{m}}\)
• Goal: Get T by itself

• First, eliminate the fraction by multiplying both sides by m:
  • Left side: \(\mathrm{m \times v^2 = mv^2}\)
  • Right side: \(\mathrm{m \times (\frac{LT}{m}) = LT}\)
  • Result: \(\mathrm{mv^2 = LT}\)

• Next, isolate T by dividing both sides by L:
  • Left side: \(\mathrm{\frac{mv^2}{L}}\)
  • Right side: \(\mathrm{\frac{LT}{L} = T}\)
  • Result: \(\mathrm{T = \frac{mv^2}{L}}\)

Answer: A. \(\mathrm{T = \frac{mv^2}{L}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand what "express T in terms of..." means, leading them to randomly manipulate the equation without a clear goal. They might try to solve for v instead of T, or get confused about which variable should be isolated.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the goal but make algebraic errors. Common mistakes include:

• Forgetting to multiply/divide BOTH sides of the equation
• Incorrectly handling the fraction LT/m when multiplying by m
• Getting the order of operations wrong

For example, they might incorrectly get \(\mathrm{T = mLv^2}\) or \(\mathrm{T = \frac{L}{mv^2}}\), which could lead them to select Choice C (\(\mathrm{T = \frac{mL}{v^2}}\)) or Choice D (\(\mathrm{T = \frac{L}{mv^2}}\)).

The Bottom Line:

This problem tests fundamental algebra skills - understanding what it means to "solve for" a variable and executing the steps correctly. Success requires both clear comprehension of the goal and careful algebraic manipulation.