A 1.00 kg particle has the xy coordinates (-1.20 m, 0.500 m) and a 3.00 kg particle has the xy coordinates (0.600 m, -0.750 m). Both lie on a horizontal plane. At what (a) x and (b) y coordinates must you place a 1.00 kg particle such that the center of mass of the three-particle system has the coordinates (-0.500 m, -0.700 m)?
To find the x-coordinate and y-coordinate where you must place a 1.00 kg particle to have the center of mass at a specific position, you can use the concept of the center of mass equation. The equation can be given as:
x_CM = (m1*x1 + m2*x2 + m3*x3) / (m1 + m2 + m3)
y_CM = (m1*y1 + m2*y2 + m3*y3) / (m1 + m2 + m3)
Where:
- x_CM and y_CM are the x-coordinate and y-coordinate of the center of mass respectively.
- m1, m2, and m3 are the masses of the particles.
- x1, x2, and x3 are the x-coordinates of the particles.
- y1, y2, and y3 are the y-coordinates of the particles.
Given:
Particle 1: m1 = 1.00 kg, x1 = -1.20 m, y1 = 0.500 m
Particle 2: m2 = 3.00 kg, x2 = 0.600 m, y2 = -0.750 m
Center of mass: x_CM = -0.500 m, y_CM = -0.700 m
Let's calculate the x-coordinate and y-coordinate for the new particle:
(a) x-coordinate:
x_CM = (m1*x1 + m2*x2 + m3*x3) / (m1 + m2 + m3)
Rearranging the equation, we get:
x3 = (x_CM * (m1 + m2 + m3) - m1*x1 - m2*x2) / m3
Substituting the given values:
x3 = (-0.500 * (1.00 + 3.00 + 1.00) - 1.00*(-1.20) - 3.00*(0.600)) / 1.00
Simplifying the equation:
x3 = (-0.500 * 5.00 + 1.20 + 1.80) / 1.00
x3 = (-2.50 + 3.00) / 1.00
x3 = 0.50 m
Therefore, you must place the 1.00 kg particle at the x-coordinate of 0.50 m to have the center of mass at -0.500 m.
(b) y-coordinate:
The process for calculating the y-coordinate is similar.
y_CM = (m1*y1 + m2*y2 + m3*y3) / (m1 + m2 + m3)
Rearranging and substituting the given values:
y3 = (y_CM * (m1 + m2 + m3) - m1*y1 - m2*y2) / m3
y3 = (-0.700 * (1.00 + 3.00 + 1.00) - 1.00*(0.500) - 3.00*(-0.750)) / 1.00
y3 = (-0.700 * 5.00 - 0.500 + 2.250) / 1.00
y3 = (-3.500 - 0.500 + 2.250) / 1.00
y3 = -1.750 / 1.00
y3 = -1.750 m
Therefore, you must place the 1.00 kg particle at the y-coordinate of -1.750 m to have the center of mass at -0.700 m.
So, the x-coordinate and y-coordinate for the new particle are (a) 0.50 m and (b) -1.750 m, respectively.