Question

area plate has an 1) Find the center of mass of the plate shown. The density of 10g/cm? NY .(4,3) Recall that Sxp Lx (x) dx X= ,and Mass 7 - Syp Ly cody Mass MacBook Pro 000 80 FJ od F7 DIN FB F4 F5 F6 F9 F10 $ % A ** & 7 4 5 6 8 9 0 E R T Y O
could you please help me how to figure out this one？ And I need all work

Answer 1

The coordinates of the centre of mass (ī,y) are then :-/*(PC) oxi)dx ASį (IF(<P – [962 1 [g(x)]-)dx Where A is the area of the region bounded between the functions y = f(x) and y = g(x) in the interval x = a and x = b. f(x)-9(x) dx In our case we need to find the centre of mass of the triangle. The given triangle is the area under the straight line joining between the points (0,0) and (4,3) from x = 0 to x = 4. The coordinates of centre of mass is not affected by the density. So, we don't need to use the density given i.e. 10g/cm². First, we need to find the equation of the straight line. The equation of straight line joining two points can be written as (Yu-Yo y-yo = (x - Xo) - x0 Where (Xo. Yo) and (x1, y.) are the two points. So, the equation of straight line joining (0,0) and (4,3) will be (3-0) y-0 = (x-0) (4-0) C. FLO E 3 y = 4* So, the given triangle is the area bounded under the straight line y = -x from x = 0 to = 4. The area is bounded between the straight line and x axis So, the area will be A= Lºcx) = g(x)) dx = [*0 - 0)dx = S** x dx = LET 3 16 = -X = 6 Now we can find the x coordinate of the centre of mass 113 i= AS*<««– 9()dx=51, +(x)dx= 1 43 -x- dx = 16 8 = х

So, the x-coordinate of the centre of mass will be 8 y-coordinate will be 1 p1 1 -4 9 ý=7 Ā Sºz ([()]? – [g(x)]dx = [*3* (x)** dx = 3X3 16+ dx 0 х = 1 64* Jo 1 1 3 3 = 2X2816 So, the y-coordinate of the centre of mass will be y = 1 So, the coordinates for the centre of mass of the given region will be 8 (x,y) 3 =