How Does a Circular Hole Affect the Center of Mass of a Rectangular Plate?

[whereisccguys]

A thin rectangular plate of uniform areal density 2.96 kg/m2 has length 42.0 cm and width 26.0 cm. The lower left hand corner is located at the origin, (x,y)= (0,0) and the length is along the x-axis. There is a circular hole of radius 5.50 cm with center at (x,y)=(15.00,11.00) cm in the plate.

Calculate the distance of the plate's center of mass from the origin.

this was a 2 part questions, first part was find the mass of the plate which i got was .295 kg... the second part is calculate the distance of the plate's center of mass from the origin... my attempt to solve this problem was to divide the plate into three pieces of the x direction and then get each mass, it's distance on X, multiple each piece of mass by it's distance from X and add them up and divide by the total mass of .295 kg... and then i do the same for the Y direction... whcih gives me a center of mass at (.2984, .1756) in meters and finally givin me a distance of .3462 m from the point of origin... but this gives me the wrong answer... anyone knoe what i did wrong and if there is another way to do it?

 

[Curious3141]

A thin rectangular plate of uniform areal density 2.96 kg/m2 has length 42.0 cm and width 26.0 cm. The lower left hand corner is located at the origin, (x,y)= (0,0) and the length is along the x-axis. There is a circular hole of radius 5.50 cm with center at (x,y)=(15.00,11.00) cm in the plate.

Calculate the distance of the plate's center of mass from the origin.

this was a 2 part questions, first part was find the mass of the plate which i got was .295 kg... the second part is calculate the distance of the plate's center of mass from the origin... my attempt to solve this problem was to divide the plate into three pieces of the x direction and then get each mass, it's distance on X, multiple each piece of mass by it's distance from X and add them up and divide by the total mass of .295 kg... and then i do the same for the Y direction... whcih gives me a center of mass at (.2984, .1756) in meters and finally givin me a distance of .3462 m from the point of origin... but this gives me the wrong answer... anyone knoe what i did wrong and if there is another way to do it?

The simplest way to do this problem is to first assume that the circular hole isn't there, find the position of the center of mass of the whole rectangular plate (it'll be in the center of the rectangle), then find the center of mass of the circular hole (center of the circle), and finally add the first mass moments of the two, treating the rectangle as a usual body BUT the circular hole as a NEGATIVE mass. You can either work with x and y coordinates separately, or just do it in column vectors and get the answer straightaway.

 

[thepatientmental]

Your approach to dividing the plate into three pieces and calculating the center of mass for each piece is correct. However, when calculating the distance of the plate's center of mass from the origin, you need to use the distance formula, which is the square root of the sum of the squares of the x and y distances. In this case, the distance formula would be:

distance = √[(0.2984)^2 + (0.1756)^2] = 0.3458 m

This is slightly different from your answer of 0.3462 m, but it may just be a rounding error. Make sure you are using the correct number of significant figures in your calculations.

Another way to approach this problem is to use the formula for the center of mass of a thin rectangular plate, which is:

xcm = (1/2)(a+b) and ycm = (1/2)(c+d)

Where a and b are the x-coordinates of the opposite corners of the plate, and c and d are the y-coordinates of the opposite corners. In this case, a=0, b=0.42 m, c=0, and d=0.26 m. Plugging these values into the formula, we get:

xcm = (1/2)(0 + 0.42) = 0.21 m and ycm = (1/2)(0 + 0.26) = 0.13 m

Using the distance formula, we get the same result as before:

distance = √[(0.21)^2 + (0.13)^2] = 0.3458 m

So, either approach should give you the correct answer. Just make sure you are using the correct formula and rounding properly.