Waves/Hooke's Law Problem

  • #1
DivGradCurl
372
0
Hi,

Could you please give a hand with this question from the book "Fundamentals of Physics/Halliday, Resnick, Walker" - 6th ed, page 395, #22P. Here it goes:

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The type of rubber band used inside some baseballs and golf balls obeys Hooke's Law over a wide range of elongation of the band. A segment of this material has an ustretched length l and mass m. When a force F is applied, the band stretches an additional length Δl.

(a) What is the speed (in terms of m, Δl, and spring constant k) of transverse waves on this stretched rubber band?

(b) Using your answer to (a), show that the time required for a transverse pulse to travel the length of the rubber band is proportional to 1/&radic;(&Delta;l) if &Delta;l << l and constant if &Delta;l >> l

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Comments:

(a) v = &radic;([tau]/[mu]), where [tau] = F = k&Delta;l and [mu] = m/(l+&Delta;l). This gives: v = &radic;{[k&Delta;l(l+&Delta;l)]/m} --- This should be right.

(b) I'm not sure what happens in either case... my guess is that when:

&Delta;l << l, we have: v = &radic;[(kl)/m].

v = dl/dt Then: [&int;(0,&Delta;l)] dt = [&int;(0,&Delta;l)] (1/v) dl

This gives: t = [2&radic;(&Delta;l)]/&radic;(k;m)

That doesn't seem to fit, but could be close.

&Delta;l >> l, we have: v = &radic;{[k(&Delta;l)^2]/m}.

v = dl/dt Then: [&int;(0,&Delta;l)] dt = [&int;(0,&Delta;l)] (1/v) dl

This gives: t = [1/&radic;(k;m)]

That doesn't seem to fit, but could be close as well.

Thanks a lot!
 

Answers and Replies

  • #2
DivGradCurl
372
0
Just a correction

Over comment for (b) [squ](k;m) = [squ](k/m)
 

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