Center of Mass

Ms. Mikaela Irene Fudolig

Physics 71 Lecture

Motion of the Center of Mass

A point particle undergoing projectile motion

Motion of the Center of Mass

An extended object undergoing projectile motion

Center of Mass

In a system of particles, the center of mass is the point that moves as ifall mass of the system were concentrated at

that pointall external forces acting on the system were

acting on that point

Center of Mass

The motion of any system of particles can be described in terms of the motion of the center of mass and the motion of the individual particles with respect to the center of mass.

Finding the center of mass

The location of the center of mass of a system of N particles is given by:

1 1 2 2

1 2

...

...N N

cmN

m r m r m rr

m m m

i ii

cmi

i

m rr

m

Example 1

A 24-kg child is 20m from an 86-kg adult. Where is the center of mass of this system?

Exercise 1

Y&F8.44. Three odd-shaped blocks of chocolate have the following masses and center-of-mass coordinates:0.300kg, (0.200m,0.300m)0.400kg, (0.100m,-0.400m)0.200kg, (-0.300m,0.600m)

Find the coordinates of the center of mass of the system.

Finding the Center of Mass

For an extended object:

cm

rdmr

M

For a symmetric extended object with constant density, the center of mass is located at its geometric center.

Finding the Center of Mass

The center of mass of two extended objects is the same as the center of mass of two point particles located at the centers of mass of the two extended objects.

Motion of the Center of Mass

i ii

cmi

i

m vv

m

,i i

net exticm

i ii i

m aF

am m

Example 2

Jose (mJ=90.0kg) and Ramon (mR=60.0kg) are standing 20.0m apart on the slippery surface of a frozen pond. Midway between the two men, a mug sits on the ice. They pull on the ends of a light rope that is stretched between them.When Jose has moved 6.0m towards the mug, how

far and in what direction has Ramon moved? (Ans. 9.0m towards the mug)

Example 3

A cannonball of mass 2m is shot with speed v0 at an angle above the horizontal. At the highest point, it breaks into two fragments of equal mass. One fragment falls straight to the ground. How far horizontally from the launching point will the other fragment land? (The initial and final elevations are the same.)