Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

This the question given in my textbook

A thin, uniform rod of length $l$ is rotated with constant angular speed $\omega$ about an axis passing through it's one end and perpendicular to one end of the rod. Find the increase in the length. $[$ Density of material is $d$ and Young's modulus is $Y$ $]$


$-dT = dm\omega^2x$

$-\int dT ${upper limit T(x) lower limit (0)} = $\int \frac{M}{L}dx\omega^2x${upper limit x lower limit x=l ]

$T(x) = \frac{dA\omega^2L^2}{2}(1 - \frac{x^2}{L^2})$

[A=area of cross section of the rod which is constant for the rod] For element

$\frac{T(x)}{A} = Y\frac{dl}{dx}$


$dl = \frac{T(x)dx}{AY} = \frac{d\omega^2L^2}{2Y}(1 - \frac{x^2}{L^2})$

I don't understand this step $\frac{T(x)}{A} = Y\frac{dl}{dx}$ can anyone explain what is happening here thank's


EDIT== I don't know how to set upper limit and lower limit in the integration that's why i have written it like this

share|cite|improve this question
up vote 1 down vote accepted

That equation is basically equating Young's modulus to stress by strain ratio, i.e.

$$Y=\frac{T/A}{\Delta l/l}$$

  • As the tension is a function of $x$ on the rod, it is denoted as $T(x)$.
  • Area of cross section is $A$.
  • $\Delta l$, the change in length is denoted by $dl$.
  • $l$, the length under consideration is $dx$. Note that the textbook has considered an - infinitesimal length because tension $T(x)$ is not constant throughout the length of the rod. But we can assume it to be constant for an infinitesimally small length $dx$.

Also note that $dl$ is the change in length of only the $dx$ part, and not the whole rod.

Then you get your equation $$Y\frac{dl}{dx}=\frac {T(x)}A$$ Which gives the integral $$\int dl = \int_{x=0}^{x=l} \frac{T(x)dx}{YA}$$

You put limits as $x=0$ to $x=l$ because $x$ varies from $0\to l$. The LHS gives you $\Delta l$ which is the total increment in length of the whole rod.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.