Which formula gives the x-coordinate of the center of mass for multiple point masses?

Which formula gives the x-coordinate of the center of mass for multiple point masses?

• A. x_cm = (m1*x1 + m2*x2 + ... + mn*xn) / (m1 + m2 + ... + mn)
• B. x_cm = (x1 + x2 + ... + xn) / n
• C. x_cm = (m1 + m2 + ... + mn) / (x1 + x2 + ... + xn)
• D. x_cm = (m1*x1 - m2*x2 + ... + (-1)^n mn*xn) / (m1 + m2 + ... + mn)

Answer: (B)

The x-coordinate of the center of mass is found by weighting each position by its mass and then dividing by the total mass. In other words, x_cm = (m1*x1 + m2*x2 + ... + mn*xn) / (m1 + m2 + ... + mn). Heavier masses pull the center closer to their x positions, so their contributions matter more in the average. If all masses were the same, this reduces to the simple average x_cm = (x1 + x2 + ... + xn)/n, which explains why that form might look familiar. But without equal masses, the plain average doesn’t correctly locate the balance point. The other expressions don’t implement this mass-weighted averaging, so they don’t generally give the correct center of mass.

The x-coordinate of the center of mass is found by weighting each position by its mass and then dividing by the total mass. In other words, x_cm = (m1x1 + m2x2 + ... + mn*xn) / (m1 + m2 + ... + mn). Heavier masses pull the center closer to their x positions, so their contributions matter more in the average. If all masses were the same, this reduces to the simple average x_cm = (x1 + x2 + ... + xn)/n, which explains why that form might look familiar. But without equal masses, the plain average doesn’t correctly locate the balance point. The other expressions don’t implement this mass-weighted averaging, so they don’t generally give the correct center of mass.