Experimental Physics - Mechanics - Static Equilibrium 1

Experimental Physics EP1 MECHANICS

- Static Equilibrium -

Rustem Valiullin

https://bloch.physgeo.uni-leipzig.de/amr/

Experimental Physics - Mechanics - Static Equilibrium 2

O

r! iF!

Conditions for equilibrium

FdrFFr ==´= qt sin!!!

0=netF!F

!

O

r!

F!'r!

The net external force must be zero

0=nett!The net external torque about any axis must be zero

'r!

'O'OOr

!

å ==i

inet FF 0!!

å ´=i

iiOOnet Fr!!!

,t

å ´=i

iiOnet Fr!!! '',t ..'..' 11 +´++´= ii FrFr

!!!!

..)(..)( '1'1 +´-++´-= iOOiOO FrrFrr!!!!!!

åå ´-÷ø

öçè

æ ´=i

iOOii

i FrFr!!!!

' Onet ,t!=

If an object is in translational equilibrium and the net torque iszero about some point, then the net torque is zero about any point.

Experimental Physics - Mechanics - Static Equilibrium 3

Center of gravity

gm!

CM

CM

òò

åå ==

dm

dmrmmr

ri

iiCM

!!!

O

gmi

Mg

gmrMgrgmr iiCGiCG åå ==!!!

If g is constant over an object, then the center of massand the center of gravity of the object do coincide.

Center of mass:

Experimental Physics - Mechanics - Static Equilibrium 4

The ladder problem

Mg

L

0=wµ

fµ

afN

wN

sf

0=netF!

0=nett!îíì

=-=-00

MgNfN

f

sw

0cos2

sin =- aa LMgLNw

sw fMgN == acot2

ffss Nff µ=£ max,

ffff MgNMg µaµµa 2cotcot2

£Þ=£

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5

-3,0

-2,5

-2,0

-1,5

-1,0

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

tan(a)

angle

fµa

21tan ³

°» 25cra

°-°» 4560cra

O

Experimental Physics - Mechanics - Static Equilibrium 5

The ladder climbing problem

Mg

L

0=wµ

fµ

a

fN

wN

sf

mgl

0=netF!

0=nett!îíì

=+-=-

0)(0gmMN

fN

f

sw

0cos2

sin =÷øö

çèæ +- aa mglLMgLNw

ffss Nff µ=£ max,sw fLmlMgN =÷

øö

çèæ +=2

cota

( )gmMNf fffs +== µµmax,

( )22

2 LlgmMLmlMg ff £Þ+£÷

øö

çèæ + µµ

( )gmMLmlMg f +£÷

øö

çèæ + µa2

cot

( )mLM

mLmM

l f

2tan -

+£ aµ

aµ tan: LlMm f£>>

general case

Experimental Physics - Mechanics - Static Equilibrium 6

Inclined domino set

L

23

45

l llL -

022=-

- crcr lAglLAg rr

CM

2Llcr =

0

1

aL /

21L

aL

x +=2

22L

aL

x +=2L

aLixi +=

åå

å=

=

=úûù

êëé +==

n

in

ii

n

iii

nCML

aLi

nm

xmX

1

1

1, 2

1

( )aLnLnn

aL

nLn

nX nCM 2

)1(22

112

1, ++=++»

LX nCM £,an £+156 =Þ= na

The sixth domino will crush the system.

5

Experimental Physics - Mechanics - Static Equilibrium 7

Ø The two conditions have to be fulfilled for a body to be

in static equilibrium:

- the net external force must be zero;

- the net external torque must be zero.

Ø If an object is in static equilibrium under action of

three non-parallel forces, the lines of action of these

forces must intersect at one point.

Ø The force of gravity can be replaced by

single force acting at the center of gravity.

To remember!