Find the center of mass of a rectangular plate of length 28.6cm and width 14.3cm . The mass density varies linearly along the length. At one end, it is 6.79g/cm^2,at the other end, it is 35.3g/cm^2
Show steps and explanation in X and Y
To find the center of mass of a rectangular plate, we need to determine its x-coordinate and y-coordinate separately. Let's start with the x-coordinate.
Step 1: Find the x-coordinate of the center of mass.
The x-coordinate of the center of mass, denoted as x_cm, can be calculated using the formula:
x_cm = (M1 * x1 + M2 * x2) / (M1 + M2)
where M1 and M2 are the masses on different parts of the plate, and x1 and x2 are the corresponding x-coordinates. In this case, we can divide the plate into two parts: the left part and the right part.
Step 2: Find the masses of the left and right parts.
The mass of each part can be calculated using the mass density and area. The mass, denoted as m, is given by:
m = density * area
where density is in g/cm^2 and area is in cm^2. For the left part, the density is 6.79 g/cm^2.
For the right part, the density is 35.3 g/cm^2.
The area of the left part is the product of the length and width of the left part. The area of the right part is the product of the length and width of the right part.
Step 3: Find the x-coordinates of the left and right parts.
The x-coordinate of the left part is 0 since it starts from the left end of the plate. The x-coordinate of the right part is the length of the plate, which is 28.6 cm.
Step 4: Substitute the values into the formula.
Now we substitute the values we found into the formula to find the x-coordinate of the center of mass:
x_cm = (M1 * x1 + M2 * x2) / (M1 + M2) = (m1 * x1 + m2 * x2) / (m1 + m2)
Substituting m1 = density_left * area_left, m2 = density_right * area_right, x1 = 0, x2 = length of plate, and simplifying, we get:
x_cm = (density_left * area_left * x1 + density_right * area_right * x2) / (density_left * area_left + density_right * area_right)
Let's calculate the values now.
density_left = 6.79 g/cm^2
density_right = 35.3 g/cm^2
area_left = length_left * width = 14.3 cm * 14.3 cm
area_right = length_right * width = 14.3 cm * 14.3 cm
length_left = 0 to length_right = 28.6 cm
Now we can substitute these values into the formula to find the x-coordinate of the center of mass.