Potential and field of a thing circular ring

[richyw]

Homework Statement

I'm trying to work through an example in my classical mechanics textbook (Fowles and Cassiday, 7th ed, example 6.7.2. The problem I am having is when he expands the integrand into a power series. I'll write out the first part of the solution. The question is "find the potential function and the gravitational field intensity in the plane of a thin circular ring

Homework Equations

$$\Phi=-G\int\frac{dM}{s}$$

The Attempt at a Solution

$$\Phi=-G\int\frac{dM}{s}=-G\int^{2\pi}_0\frac{\mu R d\theta}{s}$$where $\mu$ is the linear mass density of the ring,R is the radius of the ring and M is the mass of the ring. Then using the law of
cosines we have$$\Phi=-2R\mu G\int^{\pi}_0\frac{d\theta}{\sqrt{r^2+R^2-2Rr\cos\theta}}$$$$\Phi=-\frac{2R\mu G}{r}\int^{\pi}_0\frac{d\theta}{\sqrt{1+(R/r)^2-2(R/r)\cos\theta}}$$The next part is where I am getting stuck. It says to expand in a power series of x=R/r $$\Phi=-2x \mu G\int^{\pi}_0\left[\left( 1-\frac{1}{2}x^2+x\cos\theta\right)+\frac{3}{8}\left(x^2-2x\cos\theta\right)^2+\dots\right]d\theta$$What is happening in this step? Sorry if this belongs in intro physics. It's a junior year course though.

 

[TSny]

It's just a Taylor series expansion of $1/\sqrt{1+x}$.

See line # 4 under "Not so elementary forms" here: http://web.mit.edu/kenta/www/three/taylor.html

 

[richyw]

oh my goodness. thank you.