Center Of Mass

Center of MassCenter of Mass

Image: http://oregonstate.edu/instruct/exss323/Lecture_06.pdf

The center of mass of a body or a The center of mass of a body or a system system

of bodies is the point that moves as of bodies is the point that moves as

though all of the though all of the

mass were mass were

concentrated there concentrated there

and all external and all external

forces were forces were

applied there.applied there.

Motion of the Center of Motion of the Center of MassMass

See animations of projectile motion of See animations of projectile motion of rotating and non-rotating objects at:rotating and non-rotating objects at:

http://www.kettering.edu/~drussell/Dehttp://www.kettering.edu/~drussell/Demos/COM/com-a.htmlmos/COM/com-a.html

Influences of Body Influences of Body PositionPosition

Can use changes in body position to:Can use changes in body position to: Increase take-off height of COM (raise Increase take-off height of COM (raise

arms)arms) Decrease landing height (lift legs)Decrease landing height (lift legs) Increase height of individual body parts Increase height of individual body parts

during flight (lower other parts)during flight (lower other parts)

http://oregonstate.edu/instruct/exss323/Lecture_06.pdf

Center of Mass MotionCenter of Mass Motion

See animated video of a hammer See animated video of a hammer thrown.thrown.

Watch the motion of the center of Watch the motion of the center of mass:mass:

http://www.regentsprep.org/Regents/phttp://www.regentsprep.org/Regents/physics/phys06/acentomas/default.htmhysics/phys06/acentomas/default.htm

High JumpHigh Jump

Trajectory of the center of mass is Trajectory of the center of mass is determined when jumper leaves determined when jumper leaves ground (including maximum height ground (including maximum height of COM)of COM)

Jumper changes body position in Jumper changes body position in midair to improve performancemidair to improve performance

http://oregonstate.edu/instruct/exss323/Lecture_06.pdf

Center of Mass EquationCenter of Mass Equation

For two masses m1 and m2, the center of For two masses m1 and m2, the center of mass is at:mass is at:

21

2211

mm

xmxmxCM

Center of Mass EquationCenter of Mass Equation

For many particles, For many particles, the center of mass the center of mass can be written as:can be written as:

n

ii

n

iii

cm

m

xmx

1

1

1-D Center of Mass 1-D Center of Mass exerciseexercise

Find the center of Find the center of mass of three mass of three particles:particles:

1 kg 2 kg4 kg

mmkgkgkg

mkgmkgmkgxcm 57.2

7

18

241

)5)(2()2)(4()0)(1(

Center of Mass 3-DCenter of Mass 3-D

In 3 dimensions the same equations apply:In 3 dimensions the same equations apply:

n

ii

n

iii

cm

m

xmx

1

1

n

ii

n

iii

cm

m

ymy

1

1

n

ii

n

iii

cm

m

zmz

1

1

2-D exercise2-D exerciseFind the center of mass of a Find the center of mass of a

system of three particles:system of three particles:

ParticleParticle Mass Mass (kg)(kg) x (cm)x (cm) y (cm)y (cm)

11 1.21.2 00 00

22 2.52.5 140140 00

33 3.43.4 7070 121121

1 2

3

Answer to 2-D exerciseAnswer to 2-D exercise

cm

kgkgkg

cmkgcmkgcmkg

mmm

xmxmxm

m

xmx n

ii

n

iii

cm

83

4.35.22.1

)70)(4.3()140)(5.2()0)(2.1(

321

332211

1

1

cm

kgkgkg

cmkgcmkgcmkg

mmm

ymymym

m

ymy n

ii

n

iii

cm

58

4.35.22.1

)121)(4.3()0)(5.2()0)(2.1(

321

332211

1

1

1 2

3

Exercise: non-uniform Exercise: non-uniform diskdisk

Find the center of mass of a disk of Find the center of mass of a disk of radius 2R from which an off-center radius 2R from which an off-center disk of radius R is missing:disk of radius R is missing:

2RR

Non-uniform diskNon-uniform disk

Consider 3 disks: small (filled), large Consider 3 disks: small (filled), large (filled), and non-symmetrical:(filled), and non-symmetrical:

2RR

DiskDisk COMCOM massmass

SmallSmall -R-R mmss

LargeLarge 00 mmLL=m=mss+m+m

NSNS

Non-Non-symsym

?? mmNSNS

Non-uniform diskNon-uniform diskThe center of mass The center of mass

of a large filled disk of a large filled disk is at the origin: is at the origin:

Solve for xSolve for xNSNS: :

0

NSs

NSNSssL mm

xmxmx

33)2(

))((2

3

22

2 R

R

R

RR

RR

m

xmx

NS

ssNS

2RR

Solid BodiesSolid Bodies

For an infinite For an infinite number of number of individual particles:individual particles:

Replace Replace summation with summation with integrals:integrals:

n

ii

n

iii

cm

m

xmx

1

1

M

xdm

dm

xdmxcm

Solid Bodies: integrateSolid Bodies: integrate Use density: Use density:

Then the integral becomes:Then the integral becomes:

We will integrate over solid objects when we We will integrate over solid objects when we get to E&Mget to E&M

dVdmdV

dm

V

M

xdVV

xdVM

dxxM

xdmM

xcm111