Deriving the first moment of area of semicircle

[Elbobo]

Homework Statement

Derive via integration the first moment of area Q of a semicircle with radius r.

Homework Equations

$$Q = \int_{A} y dA$$

$$A_{semicircle} = \frac{\pi r^{2} }{2}$$

The Attempt at a Solution

$$A = \frac{\pi r^{2} }{2}$$
$$A(y) = \frac{\pi y^{2} }{2}$$
$$dA = \pi y dy$$

$$Q = \int^{y=r}_{y=0} y dA$$
$$= \int^{r}_{0} \pi y^{2} dy$$
$$= \frac{\pi}{3} [y^{3}]^{r}_{0}$$

$$Q = \frac{\pi r^{3}}{3}$$But the answer is $$\frac{2 r^{3} }{3}$$, which my textbook derived from the equation $$Q = (area) \times (centroidal height)$$. I want to know how to derive the Q for any shape without knowing its centroidal height beforehand. Can someone help me out with why I got a different and wrong answer?

 

[nvn]

Elbobo: dA is not pi*y*dy. Hint: Shouldn't dA instead be, dA = 2[(r^2 - y^2)^0.5]*dy? Try again.

 

[Elbobo]

Sorry, I really don't understand why dA equals that. My A(y) must be wrong then? What should it be and why?

 

[nvn]

Elbobo: A(y) = integral(dA), integrated from y = y1 to y = r. In your particular case, y1 = 0.

 

[DeeNos]

Your solution is almost correct, but you made a small error in your integration. The correct integral should be:

Q = \int^{y=r}_{y=0} y dA
= \int^{r}_{0} \pi y \cdot \pi y dy
= \frac{\pi^{2}}{2} [y^{3}]^{r}_{0}

Q = \frac{\pi^{2} r^{3}}{2}

This is still not the same as the expected answer of \frac{2 r^{3} }{3}. The reason for this is that you are using the wrong method to find the first moment of area for a semicircle. The correct method is to use the equation Q = (area) \times (centroidal height) as your textbook did.

The reason for this is that the first moment of area is a measure of the distribution of an object's area around a given axis. In the case of a semicircle, the centroidal height is the distance from the axis of rotation to the centroid of the semicircle. This is not the same as the average height of the semicircle, which is what you calculated in your solution.

To understand this better, imagine a semicircle with radius r and a centroidal height of h. The area of this semicircle is \frac{\pi r^{2} }{2}, and the centroidal height is given by the formula h = \frac{4r}{3\pi}. If you plug this into the equation Q = (area) \times (centroidal height), you will get the correct answer of \frac{2 r^{3} }{3}.

In summary, your solution is incorrect because you used the wrong method to find the first moment of area for a semicircle. The correct method is to use the equation Q = (area) \times (centroidal height), which takes into account the distribution of area around the axis of rotation.