Respuesta :
Answer:
x = -1.20 m
y = -1.12 m
Explanation:
as we know that four masses and their position is given as
5.0 kg (0, 0)
2.9 kg (0, 3.2)
4 kg (2.5, 0)
8.3 kg (x, y)
As we know that the formula of center of gravity is given as
[tex]x_{cm} = \frac{m_1 x_1 + m_2x_2 + m_3x_3 + m_4x_4}{m_1 + m_2 + m_3 + m_4}[/tex]
[tex] 0 = \frac{5(0) + 2.9(0) + 4(2.5) + 8.3 x}{5 + 2.9 + 4 + 8.3}[/tex]
[tex]10 + 8.3 x = 0[/tex]
[tex]x = -1.20 m[/tex]
Similarly for y direction we have
[tex]y_{cm} = \frac{m_1 y_1 + m_2y_2 + m_3y_3 + m_4y_4}{m_1 + m_2 + m_3 + m_4}[/tex]
[tex] 0 = \frac{5(0) + 2.9(3.2) + 4(0) + 8.3 y}{5 + 2.9 + 4 + 8.3}[/tex]
[tex]9.28 + 8.3 x = 0[/tex]
[tex]x = -1.12 m[/tex]
The location of the fourth object such the center of gravity of the four-object arrangement will be at (0.0, 0.0) m is;
(-1.20, -1.12) m
We are given the mass distribution as;
5kg at (0, 0)
2.9 kg at (0, 3.2)
4 kg at (2.5, 0)
8.3 kg at say (x, y)
Now, to get the position of where the fourth object will be placed so that the centre of gravity will be at (0, 0), we will use the center of gravity formula. Thus;
x' = [(m1*x1) + (m2*x2) + (m3*x3) + (m4*x4)]/(m1 + m2 + m3 + m4)
Plugging in the relevant values gives;
x' = [(5*0) + (2.9*0) + (4*2.5) + (8*x)]/(5 + 2.9 + 4 + 8.3)
x' = (8.3x + 10)/20.2
Similarly;
y' = [(m1*y1) + (m2*y2) + (m3*y3) + (m4*y4)]/(m1 + m2 + m3 + m4)
Plugging in the relevant values gives;
y' = [(5*0) + (2.9*3.2) + (4*0) + (8.3*y)]/(5 + 2.9 + 4 + 8.3)
y' = (9.28 + 8.3y)/20.2
For the center of gravity to be at (0,0), then;
x' = 0 and y' = 0. Thus;
(8.3x + 10)/20.2 = 0
8.3x = -10
x = -10/8.3
x = -1.2 m
Likewise;
(9.28 + 8.3y)/20.2 = 0
8.3y = -9.28
y = -9.28/8.3
y = -1.12 m
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