1
$\begingroup$

I'm sorry if this question is not at the right place, I didn't know if I should post it in the math or physics stackexchange...
Anyways, I encountered a step in a derivation that I can't really figure out. I'm trying to figure out the potential energy of a wave on a string and it starts like this:

To a good approximation, the extended length of a string segment $\delta s$ is related to the unstretched length $\delta x$

$\delta s = \frac{\delta x}{\cos{\theta}} = \frac{\delta x}{(1-\sin^2\theta)^{1/2}}$
Since $\theta $ is small,

$\delta s \simeq \frac{\delta x}{(1-\theta^2)^{1/2}} \simeq \delta x (1+\frac{1}{2}\theta^2)$
This last step is the thing I don't get... Is it some kind of authorized approximation from the small angle? I don't see where the square root goes or how the one half pops out, but this is what is witten in my text book...
Also sorry hat I bring small angle approximations to you, the mathematicians, I've heard you are not very big fans of them ;) I thank you in advance for your help!

$\endgroup$
1
$\begingroup$

We have that by Binomial approximation for $x$ small

$$(1+x)^r \approx 1+rx$$

here $r=-\frac12$ and then

$$\delta s \simeq \frac{\delta x}{(1-\theta^2)^{1/2}} = \delta x(1-\theta^2)^{-1/2} \simeq\delta x \left(1+\frac{1}{2}\theta^2\right)$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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