Center of Mass Powerpoint

5/28/2018 Center of Mass Powerpoint

1/24

Center of Mass

AP Physics C

Mrs. Coyle


2/24

Center of Mass

The point of an object at which allthe mass of the object is thought tobe concentrated.

Average location of mass.


3/24

Experimental Determinationof CM

Suspend the object from twodifferent points of the object.

Where two vertical lines from thesetwo points intersect is the CM.


4/24

Location of Center of Mass

The CM could be located:

within the object (human

standing straight)

outside the object (high jumperas she goes over the bar)


5/24

Center of Mass is outside the

object.


6/24

Center of Gravity

The point of the object where theforce of gravity is thought to beacting.

Average location of weight.

If g is the same throughout theobject then the CM coincides with theCG.


7/24

Center of Massof:

System of Particles

Extended Object


8/24

Center of Mass of a System ofParticles in one Dimension

XCM= mxM

miis the mass of each particle

xi is the position of each particlewith respect to the origin

M is the sum of the masses of all

particles


9/24


10/24

Example 1: Center of Massin one Dimension

Find the CM of a system of fourparticles that have a mass of 2 kgeach. Two are located 3cm and 5 cm

from the origin on the + x-axis andtwo are 2 and 4 cm from the originon the x-axis

Answer: 0.5cm


11/24

Coordinates of Center ofMass of a System of

Particles in ThreeDimensions

CM CM CM

i i i i i i

i i i

m x m y m z

x y z

M M M


12/24

Coordinate of CM using thePosition Vector, r

CM

i i

i

m

M

rr

i i i i

x y z r i j k


13/24

Example 2: Center of Massin two Dimensions

Find the CM of the following system:

Ans: x=1m, y=0.33m

m1=1kg

m2=2kg m3=3kg2m

2m


14/24

Center of Mass of anExtended Object

An extendedobject can beconsidered adistribution ofsmall masselements, Dm.


15/24

Center of Mass of anExtended Object using

Position Vector

Position of the center of mass:

CM

1dm

M

r r


16/24

Center of Mass of anExtended Object

CM CM

CM

1 1

1

x x dm y y dmM M

z z dmM


17/24

CM of Uniform Objects

Uniform density,

=m/V=dm/dV

Uniform mass per unit length,

= m/x = dm/dx


18/24

Center of Mass of a Rod Find the center of mass of a rod of mass

Mand length L.

Ans

:x

CM=L/ 2, (or

yCM =

zCM = 0)


19/24

CM of Symmetrical Object

The CM of any symmetrical

object lies on an axis ofsymmetry and on any plane ofsymmetry.


20/24

Toppling Rule of Thumb

If the CG of theobject is above thearea of support, the

object will remainupright.

If the CG is outside

the area of supportthe object will topple.


21/24

Another look at Stability

Stable equilibrium: when for abalanced object a displacementraises the CG (to higher U so it tends

to go back to the lower U).

Unstable equilibrium: when for abalanced object a displacement

lowers the CG (lower U).

Neutral equilibrium: when theheight of the CG does not change

with displacement.


22/24

Stability


23/24

Example #41A uniform piece of sheet steel is

shaped as shown. Compute the x and

y coordinates of the center of mass.

30 cm

20 cm

10 cm

20 cm 30 cm10 cm0 cm

Ans:x=11.7cm,y=13.3cm


24/24

Example #44 Fosbury Flop

Find the CM

Ans: 0.0635L belowthe top of the arch

Documents

Center of Mass Powerpoint