Wave Speed on a Stretched String — Question 9

Question 9

A guaranteed speed interval uses opposite parameter extremes

Tension and linear density have independent hard bounds $9\le F\le11\,\mathrm{N}$ and $.009\le\mu\le.011\,\mathrm{kg/m}$.

1. Find the tight speed bounds allowed by these intervals.

2. Explain why using the lower bounds together and upper bounds together would not bound the speed. Compare the central value with the range.

Question 9 - Solution

A guaranteed speed interval uses opposite parameter extremes

Method and mathematical working

The speed increases with force and decreases with density. For independent hard bounds, pair opposite endpoints to obtain the extreme ratios.$$\begin{aligned}v_{\min}&=\sqrt{F_{\min}/\mu_{\max}},\\v_{\max}&=\sqrt{F_{\max}/\mu_{\min}}.\\[4pt]v_{\min}&=\sqrt{9/0.011}=28.60\,\mathrm{m/s},\\v_{\max}&=\sqrt{11/0.009}=34.96\,\mathrm{m/s}.\end{aligned}$$

(a) Calculation and conclusion

Speed increases with $F$ and decreases with $\mu$. Thus $\sqrt{9/.011}\le v\le\sqrt{11/.009}$, or $28.60\le v\le34.96\,\mathrm{m/s}$. Independent bounds permit the crossed extreme pairs.

(b) Interpretation and limits

Pairing low with low or high with high would give the same value $\sqrt{1000}=31.62\,\mathrm{m/s}$ and miss the actual extremes. This central value is valid only for the central ratio; hard bounds require monotonic endpoint analysis, not cancellation of uncertainties.

Physical check

Pairing both lower endpoints would not minimize the ratio. These are guaranteed interval bounds under the stated independence, not a statistical uncertainty or a confidence interval.