Tension in Spinning Rod

The tension within a homogeneous spinning rod, of constant angular velocity, continuously increases as the distance from the pivot point increases. Considering this continuous change in tension what prevents the rod from continuously growing in length?
- Sam (age 17)
Boise Idaho USA

What holds the rod together is just the usual forces between atoms that hold things together. I think what you're asking is whether the lengthening that the tension causes will then lead to more force, and more lengthening, so that the rod will fly apart. That depends on how fast it's spinning and how much the resulting tension lengthens it.

For a simple picture, say that the tension lengthened the rod by 1%. Now the 1% longer rod has an extra 1% tension. So that gives another 1% of 1% lengthening. Etc. Adding it ll up the extra lenght is 0.01 +0.0001 + 0.000001... times the initial length. Summing that series, the result is just a factor of 1/0.99. 

You can see that this argument says that the lengthening won't be infinite so long as the initial factor is less than 100% lengthening. That means that if you spin the rod fast enough, it will fall apart, just as you were wondering. Arguments like this give maximum spinning rates for stars, held together by gravity, before they fall apart.

Mike W.

(published on 03/04/2020)