Locate the center of mass of three particles of mass m1=1.0kg,m2= 2.0kg, and m3= 3.0kg at the corners of an equilateral triangle 1.0m on a side
lt (0,0) be one corner where m1 is, and the x axis along the line from m1 to m3.
cg*totalmass=(0,0)*1 + (1,0)*2 + (.5,.707)3
cg=i(0+2+1.5)/6+j(0+O+2.12)/6
cg= i(3.5/6) + j(2.12/6) where i is in base direction,j is perpendicular to it,measured from one base corner
To locate the center of mass of the three particles, we need to find the coordinates (x, y) of the center of mass point. We can start by setting up a coordinate system and assigning coordinates to each of the particles.
Since the particles are located at the corners of an equilateral triangle, we can choose one corner as the origin (0, 0) and align the x-axis along the base of the triangle. Let's assume that the bottom left corner (m1) is located at (0, 0), the bottom right corner (m2) is located at (d, 0), and the top corner (m3) is located at (d/2, h). Here, d represents the length of the triangle side and h represents the height of an equilateral triangle.
Given that d = 1.0 m, and assuming h is the height of an equilateral triangle with side length d, we can calculate the value of h using the formula h = (sqrt(3)/2) * d.
Substituting the values, h = (sqrt(3)/2) * 1.0 = sqrt(3)/2.
Now, we can find the x-coordinate of the center of mass (x_cm) using the formula:
x_cm = (m1*x1 + m2*x2 + m3*x3) / (m1 + m2 + m3)
Since m1, m2, and m3 are known, let's calculate the x-coordinate:
x_cm = (1.0*0 + 2.0*d + 3.0*(d/2)) / (1.0 + 2.0 + 3.0)
Simplifying further, we get:
x_cm = (0 + 2.0*d + 1.5*d) / 6.0
= (2.0 + 1.5) * (d/6.0)
= 3.5 * (d/6.0)
Substituting the value of d, we have:
x_cm = 3.5 * (1.0/6.0) = 0.5833
Now, let's find the y-coordinate of the center of mass (y_cm) using the formula:
y_cm = (m1*y1 + m2*y2 + m3*y3) / (m1 + m2 + m3)
Again, substituting the known values:
y_cm = (1.0*0 + 2.0*0 + 3.0*h) / (1.0 + 2.0 + 3.0)
= 3.0 * (h/6.0)
= 3.0 * (sqrt(3)/12.0)
Simplifying further, we get:
y_cm = 3.0 * (sqrt(3)/12.0)
= 0.866 * (sqrt(3)/4)
= 0.2887
Therefore, the center of mass of the three particles is located at approximately (0.5833, 0.2887).