Which expression gives the y-coordinate of the center of mass for multiple point masses?

Which expression gives the y-coordinate of the center of mass for multiple point masses?

• A. y_cm = (y1 + y2 + ... + yn) / n
• B. y_cm = (m1*y1 - m2*y2 + mn*yn) / (m1 + m2 + ... + mn)
• C. y_cm = (m1*y1 + m2*y2 + ... + mn*yn) / (m1 + m2 + ... + mn)
• D. y_cm = (m1*y1 + m2*y2) / (m1 + m2)

Answer: (C)

The y-coordinate of the center of mass for a system of point masses is the mass-weighted average of their y positions. For each mass, multiply its y position by its mass, sum those products, and then divide by the total mass. This gives y_cm = (m1*y1 + m2*y2 + ... + mn*yn) / (m1 + m2 + ... + mn). Heavier masses pull the center toward their y-values, so their y coordinates contribute more to the numerator. If all masses were equal, you’d get the simple average of the y-values, but in general you need the weights to reflect the different masses. The other forms either ignore the weights, include an incorrect sign, or only apply to two masses, so they don’t correctly represent the center of mass for a system with multiple masses.

The y-coordinate of the center of mass for a system of point masses is the mass-weighted average of their y positions. For each mass, multiply its y position by its mass, sum those products, and then divide by the total mass. This gives y_cm = (m1y1 + m2y2 + ... + mn*yn) / (m1 + m2 + ... + mn). Heavier masses pull the center toward their y-values, so their y coordinates contribute more to the numerator. If all masses were equal, you’d get the simple average of the y-values, but in general you need the weights to reflect the different masses. The other forms either ignore the weights, include an incorrect sign, or only apply to two masses, so they don’t correctly represent the center of mass for a system with multiple masses.