How do I find the center of mass using polar coordinates?

[vadar]

Help! I've been trying to do this for ages, but i know there's some silly little thing I am doing wrong!
Problem, Find the center of mass of a constant density semicircular plate of radius a, using polar co-ordinates, what regions do i use for the double intergral with respect to r and θ?

 

[J77]

The angle $$\theta$$ from 0 to $$\pi$$, and the radius from 0 to a.

The dA part of the double integration becomes $$r dr d \theta$$

You assume uniform density, so you just intgrating $$r\cos\theta$$ and $$r\sin\theta$$ to get the coordinates to the centre of mass.

 

[tacman]

Don't worry, finding the center of mass using polar coordinates can be tricky at first. The first thing you need to do is set up the double integral for the center of mass formula, which is ∫∫rρ(r,θ)dA, where r represents the distance from the origin and ρ(r,θ) represents the density function.

In this case, since the plate has a constant density, ρ(r,θ) will just be a constant. Now, for the bounds of the integral, you will need to consider the semicircular region in polar coordinates. This means that r will go from 0 to a, and θ will go from 0 to π/2.

To visualize this, think of the semicircular plate as a pizza slice, with the curved edge being the arc of the circle and the straight edge being the diameter. The radius, r, will vary from 0 to the length of the radius of the circle, which is a. And the angle, θ, will vary from 0 to π/2, which is half of the full circle.

So, your double integral will look something like this:

∫0π/2 ∫0a rρ(r,θ)drdθ

Make sure to check your work and plug in the correct bounds for r and θ, as well as the correct expression for ρ(r,θ). I hope this helps and good luck with your problem!