[Math] Center of mass of the semi cylindrical shell.

Question

I have to find the x and z co-ordinates of the centre of mass.

Here is finding the x;

$$V \bar{x} = \int_{-l}^0x dV$$

$$\implies \pi r \delta r l \bar{x} = \int_{-l}^0x\pi r \delta r \delta x$$
$$\implies \pi r \delta r l \bar{x} = -( \frac{l^2}{2} \pi r \delta r) $$
$$ \bar{x} = – \frac{l}{2} $$

and if I substitute the values in, I get the correct answer.

I am having trouble with the z-coordinate. Could someone give me a hand please.

Answer 1

Consider an circular arc.

r - z = 2r/pi

z = [r (pi-2)]/ pi

Where r is the radius