POP QUIZMomentum and Collisions On loose-leaf paper answer the following questions.

Be sure to show all work and include units.

1. Compare and contrast inelastic and elastic collisions.

2. If an object is traveling with a speed of 5 m/s, at an angle 46° North of East when it explodes into three equal parts. Find the velocity of the third fragment if the other two travel at velocities of 8 m/s east and 10 m/s 30° south of west.

Center of Mass

AP Physics C

Momentum of point massesUp to the point in the course we have treated everything as a single

point in space no matter how much mass it has. We call this "Special point" the CENTER OF MASS. The CM is basically a coordinate which represents the average position of mass.

Consider an extended rigid body made up of 2 or more point masses, such as a dumb-bell. We want to treat this as a system, but we don't want to just assume that the CM is in the middle (even though we know it is).

The reason we are concerned here is because if this system were to move, how would you determine the momentum? Would you just assume ALL the mass was in the middle and treat it as a point?

Center of MassLet's assume that the rod that holds the two masses together is very light and that its mass is negligible compared to the weighted ends.

To calculate the "X" position of the center of mass we would do the following, which looks like taking an average.

The formal equation to find the CENTER OF MASS! Keep in mind that you can also usethis for the Y and Z coordinate!

total

N

i iiCM M

xmX

1

ExampleConsider the following masses and their coordinates

which make up a "discrete mass" rigid body.

What are the coordinates for the center of mass of this system?

Example cont’M

zmr

M

ymr

M

xmr

N

i iicm

N

i iicm

N

i iicm zyx

111

16

)10)(1()7)(10()2)(5(

16

)17)(1()2)(10()0)(5(

16

)10)(1()4)(10()3)(5(

1

1

1

M

zmr

M

ymr

M

xmr

N

i iicm

N

i iicm

N

i iicm

z

y

x

-0.94

0.19

4.4

kjircm ˆ4.4ˆ19.0ˆ94.0

The center of mass of a symmetrical shape is located on the axis or plane of symmetry. The CM of a sphere is the

center.

The center of mass of a more complex arrangements of shapes can be found by finding the CM of each partial shape, and then finding the CM of the whole system.

Center of mass of a shape with uniform density

©2008 by W.H. Freeman and Company

A continuous body?What if you had a body that had a non-uniform mass

distribution throughout its structure? Since for a rigid body we used SUMMATION, , S it only make sense that we would use INTEGRATION to sum all of the small individual masses thus allowing us to determine the center of mass of an object.

Here is the SAME equation as before yet with the appropriate calculus symbols we need. The equation is the same for the y, and z positions.

Example

t

b

y

xa

So to solve practical problems we need to define a set of spatial expressions

Linear Mass Density Surface Area Density Volume Density

These only work for UNIFORM DENSITY objects.

Steps for Using Integration to Find the Center of Mass1. Decide if the object is of uniform or non-uniform

density.

2. If the object is uniform density then we know the mass, it is M. If it is non-uniform, we do not know and we must integrate the density from 0 to L to find the mass.

3. Solve the density equation for M and set up a differential equation.

4. Substitute steps 2 and 3 into the center of mass equation and integrate over the interval 0 to L. Then you have the center of mass.

A 1-D thin uniform rod is aligned along the x-axis between x = a & x = b. The total mass of the ROD is M. Determine the X coordinate for the center of mass. We certainly can guess that the CM would be L/2, but let’s see if our method works.

a b

dxdmdx

dmMicro

ab

M

L

MMacro

L

y

x

dx

x

Non-Uniform Density

Find the center of mass of a non-uniform rod of length L and mass M whose density is given by λ=kx.

Before we begin, we must find the mass by integrating the density from 0 to L. Then we can solve it like normal!

Let’s Put Numbers to It.

A non-uniform rod of length 2 m has a density λ=kx2 where k = 10 g/m3.

Determine the Mass of the rod. Determine the Center of Mass of the rod.