Calculating the Mass of a Cycloid Wire with Constant Density | Integration Help

[kasse]

Homework Statement

What is the mass of a wire shaped like the arch x=t-sint, y=1-cost (t from 0 to 2*pi) of a cycloid C that has constant density D=k?

The Attempt at a Solution

I must integrate D*ds. I find that ds=sqrt(2-2cost)dt. Is this wrong? If it's right, I don't know how to integrate k*sqrt(2-2cost).

 

[Dick]

There is a trick. sqrt(1-cos(t)) can be written neatly in terms of sin(t/2). Check out half angle formulas.

 

[kasse]

m=8k is my answer then.

 

[kasse]

There is a trick. sqrt(1-cos(t)) can be written neatly in terms of sin(t/2). Check out half angle formulas.

I think it's supposed to be sqrt((1-cos(t))/2) that can be written in terms of sin(t/2), right?

 

[Dick]

Right. I didn't mean that they were equal - just that they were closely related.

 

[kasse]

Then I'll try to find the centroid, first the x:

Then I must integrate x*k*ds and multiply with 1/m=1/8k.

If I've done right, I'll have to integrate (t-sin(t)*sin(t/2). Is it time for another trick?

 

[Dick]

Not as tricky as the first one. t*sin(t/2) is a routine integration by parts. I would write sin(t)*sin(t/2) as 2*sin(t/2)*cos(t/2)*sin(t/2) using double angle formula and do a substitution. Nothing unusual here.