A thin rectangular plate of uniform areal density ? = 3.13 kg/m 2 has length of

Question

A thin rectangular plate of uniform areal density ? = 3.13 kg/m2 has length of 41.0 cm and width of 21.0 cm. The lower left hand corner is located at the origin, (x,y)= (0,0) and the length is along the x-axis.

-Calculate the x-coordinate of CM of the plate.

A thin rectangular plate of uniform areal density sigma = 3.13 kg/m2 has length of 41.0 cm and width of 21.0 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 7.50 cm with center at(x,y)= (14.50, 8.50) cm in the plate. Calculate the mass of plate. Calculate the x-coordinate of CM of the plate. Calculate the distance of the plate's CM from the origin.

Explanation / Answer

The area of the plate is 21.0cm * 41.0 cm = 861cm^2

The xCM of the plate is 20.5cm and the y CM is 10.5cm

The area of the hole is -(?*r^2) = -(?*7.50^2) = -176.625cm^2

and x CM of the hole is 14.50cm and y CM is 8.50 cm

Now sum the xCM*A for each body => 20.5*861 + (14.50*(-176.625) = 15089.43 cm^3

and yCM*A = 861*10.5 + 8.5*(-176.625) = 7539.1875 cm^3

Now divide each of the by the total area to find the CM of the composite figure

So A = 861 - 176.625 = 684.375cm^2

So x CM = 15089.43/684.375 = 22.048 cm

and y CM = 7539.1875/684.375 = 11.016cm

And the distance to the origin is sqrt(22.048^2 + 11.016^2) = 24.64 cm

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