First problem: Excessive busywork led you to the correct final equation, then you did the simple math wrong. plug in your m's again.
2nd problem You are looking for the stretch of the spring. Goto Hookes' Law. How does a change in tension affect the fundimental frequncy of a standing wave in a string?
Isn't the [tex]\lambda[/tex] for the third harmonic [tex]\frac{2L}{3}[/tex]? If you use hookes law, don't we need to know the spring constant k before using this information to solve this problem?
You are right about the 2L/3, although this thread "died" 9 months ago. But to answer your questions, you don't need to know the spring constant. You just need to know the proportionality between the stretch of the string, to the tension of the spring, to the speed of the wave in the string. If the frequency is constant, by doubling the stretch of the spring, the tension will double, and the wave speed will increase by 1.41 (square root 2). to have the same frequency go from being the 3rd harmonic to being the 2nd harmonic (2 is 2/3 of 3), the speed will have to increase by 3/2 of it's original, and the tension and therefore stretch will have to increase by the square root of 3/2 Hmm, but that doesn't give 18 cm. Just a minute...d'oh! the stretch will increase by the square of 3/2 (not the root).