# 0.18 Phy1160: force and motion -- momentum, impulse, and conservation  (Page 10/14)

 Page 10 / 14

## Center of mass

Interactions between parts of a system transfer momentum between the parts, but do not change the total momentum of the system. We can define apoint called the center of mass that serves as an average location of a system of parts.

The center of mass need not necessarily be at a location that is either in or on one of the parts. For example, the center of mass of a pair of heavy rods connected at oneend so as to form a "V" shape is somewhere in space between the two rods.

Having determined the center of mass for a system, we can treat the mass of the system as if it were all concentrated at the center of mass.

## Location of the center of mass

For a system composed of two masses, the center of mass lies somewhere on a line between the two masses. The center of mass is a weighted average of the positions of the twomasses.

Facts worth remembering -- Center of mass for two objects

For a pair of masses located at two points along the x-axis, we can write

xcm = (m1*x1/M) + (m2*x2/M)

where

• xcm is the x-coordinate of the center of mass
• m1 and m2 are the values of the two masses
• x1 and x2 are the locations of the two masses
• M is the sum of m1 and m2

Multiple masses in three dimensions

When we have multiple masses in three dimensions, the definition of the center of mass is somewhat more complicated.

Facts worth remembering -- Center of mass for many objects

Vector form:

rcm = sum over all i(mi*ri / M)

Component form:

xcm = sum over all i(mi*xi / M)

ycm = sum over all i(mi*yi / M)

zcm = sum over all i(mi*zi / M)

where

• Vector form
• rcm is a position vector describing the location of the center of mass
• ri are position vectors describing the locations of all the masses
• mi are masses for i=1, i=2, etc.
• Component form
• xcm, ycm, and zcm are the locations of the center of mass along 3-dimensional axes.
• mi are masses for i=1, i=2, etc.
• xi, yi, and zi are the locations of the masses along 3-dimensional axes for i=1, i=2, etc.
• M is the sum of all of the masses

## Motion of the center of mass

It can be shown that in an isolated system, the center of mass must move with constant velocity regardless of the motions of the individual particles.

It can be shown that in a non-isolated system, if a net external force acts on a system, the center of mass does not movewith constant velocity. Instead, it moves as if all the mass were concentrated there into a fictitious point particle with all the external forces acting on that point.

## Example scenarios

This section contains explanations and computations involving momentum, impulse, action and reaction, andthe conservation of momentum.

## Momentum examples

This section contains several examples involving momentum

## A sprinter

Use the Google calculator to compute the momentum of a 70-kg sprinter running 30 m/s at 0 degrees.

Answer: 2100 kg*m/s at 0 degrees

## A truck

Use the Google calculator to compute the momentum in kg*m/s of a 2205-lb truck traveling 33.6 miles per hour at 0 degrees when the changes listed belowoccur:

1. Initial momentum
2. Momentum when velocity is doubled
3. Momentum at initial velocity when mass is doubled
4. Momentum when both velocity and mass are doubled

I only see partial conversation and what's the question here!
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research.net
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