Question:A physics formula relates the time t (in seconds) for a pendulum swing to the pendulum length n (in meters)...
GMAT Advanced Math : (Adv_Math) Questions
A physics formula relates the time \(\mathrm{t}\) (in seconds) for a pendulum swing to the pendulum length \(\mathrm{n}\) (in meters) and gravitational acceleration \(\mathrm{g}\) (in m/s²): \(\mathrm{t = \sqrt{\frac{2n}{g}}}\). Which equation correctly expresses n in terms of t and g?
- \(\mathrm{n = \frac{gt}{2}}\)
- \(\mathrm{n = \frac{gt^2}{2}}\)
- \(\mathrm{n = \frac{(gt)^2}{2}}\)
- \(\mathrm{n = \frac{2t^2}{g}}\)
1. TRANSLATE the problem information
- Given: \(\mathrm{t = \sqrt{\frac{2n}{g}}}\) where t is time, n is length, g is gravitational acceleration
- Find: Express n in terms of t and g
2. INFER the approach
- Since n is inside a square root, we need to eliminate the radical first
- Strategy: Square both sides of the equation, then use algebraic manipulation to isolate n
3. SIMPLIFY by squaring both sides
- Starting equation: \(\mathrm{t = \sqrt{\frac{2n}{g}}}\)
- Square both sides: \(\mathrm{t^2 = (\sqrt{\frac{2n}{g}})^2}\)
- The square and square root cancel: \(\mathrm{t^2 = \frac{2n}{g}}\)
4. SIMPLIFY by clearing the fraction
- Multiply both sides by g: \(\mathrm{gt^2 = 2n}\)
- This eliminates g from the denominator
5. SIMPLIFY to isolate n
- Divide both sides by 2: \(\mathrm{n = \frac{gt^2}{2}}\)
Answer: B. \(\mathrm{n = \frac{gt^2}{2}}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students incorrectly square the equation, thinking that squaring \(\mathrm{t = \sqrt{\frac{2n}{g}}}\) gives \(\mathrm{t = \frac{2n}{g}}\) instead of \(\mathrm{t^2 = \frac{2n}{g}}\).
When they fail to square the left side (keeping it as just t), their subsequent algebra leads to \(\mathrm{n = \frac{gt}{2}}\).
This leads them to select Choice A (\(\mathrm{n = \frac{gt}{2}}\))
Second Most Common Error:
Poor INFER reasoning about order of operations: Students understand they need to square both sides but get confused about what exactly to square when they reach the manipulation steps.
Some students incorrectly think they should square the entire expression gt, leading them to \(\mathrm{(gt)^2}\) in the numerator instead of \(\mathrm{gt^2}\).
This may lead them to select Choice C (\(\mathrm{n = \frac{(gt)^2}{2}}\))
The Bottom Line:
This problem tests whether students can systematically work through multi-step algebraic manipulation while maintaining precision at each step. The key insight is recognizing that squaring both sides must be the first step to eliminate the radical.