Calculate the center of mass of a semi-circular metal plate

[mit_hacker]

Homework Statement

(Q) Calculate the center of mass of a semi-circular metal plate of uniform density ρ and thickness t. Let the radius of the plate be a. The mass of the plate is thus 1/2 (ρπat2). In your co-ordinate system, you must consider the x-axis passing through the bottom of the plate and the y-axis to be bisecting the metal plate.

Homework Equations

xcm = 1/M int [xdm] and ycm = 1/M int[ydm]

The Attempt at a Solution

If we take a small change in length dx, the area of the rectangle formed will be adx. The volume therefore will be atdx. The change in mass dm = ρatdx.

Substituting and integrating gives us 2a/π which is definitely wrong since under the is co-ordinate system, x-value should come to 0 as it is a uniform object. Am I right? Please help me!

 

[learningphysics]

The height of the rectangle is not a... it is $$\sqrt{a^2 - x^2}$$.

 

[m2003]

I would approach this problem by first verifying the given information and equations. The formula for calculating the center of mass of a two-dimensional object is indeed xcm = 1/M int [xdm] and ycm = 1/M int[ydm], where M is the total mass of the object and dm is the differential mass element.

Next, I would consider the geometry of the semi-circular metal plate. Since it is a two-dimensional object, we can assume that it has a negligible thickness, so we can ignore the t in the given formula for mass. The plate is also of uniform density, meaning that the density ρ is constant throughout the plate.

To find the center of mass, we need to integrate over the entire area of the plate. Since the plate is symmetric about the y-axis, we can integrate from x = -a to x = a and multiply the result by 2. This takes into account both halves of the semi-circle.

Using the given information, we can rewrite the integral as follows:

xcm = 1/M ∫ [x(ρπa^2dx)]

= 1/M ρπa^2 ∫ [xdx]

= 1/M ρπa^2 [x^2/2] from x = -a to x = a

= 1/M ρπa^2 [a^2 - (-a)^2]/2

= 1/M ρπa^2 [a^2 + a^2]/2

= 1/M ρπa^2 (2a^2)/2

= 1/M ρπa^2 (a^2)

= ρπa^2/M

Since the total mass of the plate is 1/2 (ρπa^2), we can substitute this into the equation to get:

xcm = ρπa^2/[1/2 (ρπa^2)]

= 2a/π

This matches our initial calculation, so it seems that the given equation and information are correct.

In conclusion, the center of mass of a semi-circular metal plate of uniform density and thickness t is located at 2a/π on the x-axis, where a is the radius of the plate.