Question

A uniform rectangular plate of length B = 32 cm and height A = 15 cm has a rectangular corner cut out of it of length D = 9 cm and height C = 10 cm. The plate is made of a material of area mass density σ. For this problem we set the origin at the lower left corner of the plate with the x-axis horizontal pointing right and the y-axis vertical pointing up.

(a) Calculate the value of the y-coordinate, in centimeters, for the center of mass of the plate.
(b) Calculate the value of the x-coordinate, in centimeters, for the center of mass of the plate.

Answer 1

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Step-by-step explanation:

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Answer 2

Hi there!

a)
Since the objects are made of material of a constant area/mass density, we can simplify the process.

To calculate the value of the y-coordinate for the center of mass, we can begin by finding the y-coordinates for the center of masses for both rectangles.

Also, let the small rectangle be 'Rectangle 1', and the bigger rectangle 'Rectangle 2'.

Smaller rectangle:

Since D = 9 cm and B = 32 cm, the remaining width of the small rectangle is equal to B - D = 32 - 9 = 23 cm. The midpoint of this width is:

`w_(1m) = (23)/(2) = 11.5 cm`

Now, the height of this rectangle is 10 cm. The midpoint of this height is 10/2 = 5 cm. However, this is not the actual y-coordinate. Since the height of the block below is 5 cm, we must add the two because this rectangle is on top of the other.

`h_(1m)= 5 + 5 = 10 cm`

Larger rectangle:
We can simply take the midpoints of its dimensions to solve for its center of mass.

`w_(2m) = (B)/(2) = (32)/(2) = 16 cm`

`h_(2m) = (A-C)/(2) = (15-10)/(2) = 2.5 cm`

Now, take the averages of the coordinates for both rectangles to solve.

y-coordinate:

`y_(cm) = (10 + 2.5)/(2) = (12.5)/(2) = \boxed{6.25 cm}`

x-coordinate:

`x_(cm) = (11.5 + 16)/(2) = (27.5)/(2) = \boxed{13.75 cm}`