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Course: Physics archive>Unit 6

Lesson 3: Center of mass

Equation for center of mass

The center of mass is a point in a system that responds to external forces as if the total mass of the system were concentrated at this point. The center of mass can be calculated by taking the masses you are trying to find the center of mass between and multiplying them by their positions. Then, you add these together and divide that by the sum of all the individual masses.

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• so i f you had more than 2 objects that are not in the same plane, would you just add their positions like vectors and determine the c of m from the results?
• Yes you would use the X,Y,Z coordinates......It is the unique position at which the weighted position vectors of all the parts of a system sum up to zero.....
• What about the motion of the centre of mass?
Suppose a box is moving in a projectile motion, it splits into parts somewhere in the trajectory. So, does the CoM follow the initial trajectory, or its trajectory changes?
• The COM does follow the intial trajectory, no matter what internal force was applied to split it apart. The derivation is simple: the system of the box has no new external forces, so its COM will follow the same trajectory. This is only true if the box is still in the air.
• Can someone give me a link to an article referring to how the center of mass equation was derived?
• The open source CLP 2 textbook (http://www.math.ubc.ca/~CLP/CLP2/) used at UBC has a section dedicated to deriving center of mass from torque, if you're interested. The section is 2.3.2 Optional - Torque.
• So I understand how to find the center of mass between a system of objects, but how would you find it for one single object?
• For a single object,if it is a point object then its position vector itself gives the co-ordinates of the CM.If it is an extended body like a sphere or a cube,which is symmetrical then its geometric center gives the CM of the body.
(1 vote)
• I don't get how a body's center of mass be outside it?
• try to imagine a doughnut
• If a ball strikes an object at a non-centered point, how do we find the speed of the block's rotation?
• A force applied through a point other than COM will cause a torque. Calculate the torque, and from that calculate the change in angular momentum and from that calculate the angular velocity.

You need to be a bit careful though to break up the force into components which are perpendicular and parallel to the radial vector from the COM to the point of force. The perpendicular component will generate the torque and rotational motion and the parallel component will generate a force causing linear translational motion.
• How's the center of mass gonna change if we account the weight of the string/rod connecting the two bodies?
• What is the mass of the string/rod in comparison to the mass of the two bodies? If you have a microscopically thin carbon nano-tube connecting two massed that have masses in the kilograms then there would be no measurable change in COM (Center of Mass). On the other hand if you had two styrofoam balls connected by a heavy iron chain it should make a big difference.

If you can determine the COM of the string/rod then you can calculate the the COM of the bodies and the string/rod by using both COMs.