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Center of Mass

# Center of Mass - Center of Mass The Center of Mass:Chap...

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Center of Mass

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The Center of Mass:Chap 13.2 n There is a special point in a system or object, called the center of mass , that moves as if all of the mass of the system is concentrated at that point. n The system will move as if an external force were applied to a single particle of mass M located at the center of mass n M is the total mass of the system
Center of Mass, Coordinates n Consider the system is made up of many pieces labeled with the subscript i n The coordinates of the center of mass are n where M is the total mass of the system M = m i i

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Center of Mass, position n The center of mass can be located by its position vector, r CM n r i is the position of the i th particle, defined by
C of M, Example x cm = (1.0 m )1.0 kg + (2.0 m )1.0 kg + (0 m )2.0 kg 4.0 kg x cm = 0.75 m y cm = (0 m )1.0 kg + (0 m )1.0 kg + (2.0 m )2.0 kg 4.0 kg y cm = 1.0 m r r cx = x cn ˆ i + y cm ˆ j = 0.75 m ˆ i + 1.0 m ˆ j M 1 = M 2 = 1.0kg, M 3 = 2.0kg x cm = m i x i i M

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Center of Mass, Example 2 n Both masses are on the x -axis n The center of mass is on the x -axis n The center of mass is closer to the particle with the larger mass
C of M Example 2 r r cm = m 1 x 1 + m 2 x 2 m 1 + m 2 ˆ i e . g . take m 1 = 10. kg , x 1 = 1.0 m & x 2 = 10. m , m 2 = 1.0 kg r r cm = 10. + 10. 11 ˆ i = 1.8 m ˆ i

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Center of Mass, Extended Object n Think of the extended object as a system containing a large number of particles n The particle distribution is small, so the mass can be considered a continuous mass distribution
Center of Mass, Example n An extended object can be considered a distribution of small mass elements, D m n The center of mass is located at position r CM

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Center of Mass, Extended Object, Coordinates n The coordinates of the center of mass of the object are
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