Static Equilibrium - General Physcis - Lecture Slides, Slides of Physics

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12. Static Equilibrium

1.^

Conditions for Equilibrium

2.^

Center of Gravity

3.^

Examples of Static Equilibrium

4.^

Stability

The Alamillo Bridge in Seville, is the work of architect Santiago Calatrava.What conditions must be met to ensure the stability of this dramatic?

=^

net^

net F^

τ^

Example 12.1. Drawbridge

The raised span has a mass of 11,000 kg uniformly distributed over a length of 14 m.Find the tension in the supporting cable.

= i

Force

F at hinge not known.h^ ^ Choose pivot point at hinge.

^

g^

^  T

τ^

τ^

1

2

sin^

sin^

L m g 2

L T

^

^

1

^ 

^

2

^ 

 ^

 ^

(^12) sin2sin m g T

^

^

 ^

sin

kg^ 2sin

m^

s^

^

kN ^

Another choiceof pivot: Ex 15

y

x

Tension 30 

T Gravity m

g

Hinge force

15  F h

^2 ^1

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GOT IT? 12.1.

Which pair, acting as the

only

forces on the object, results in static equilibrium?

Explain why the others don’t.



(C) (A):

F^

^^0

.

(B):

^ 

^0.

Conceptual Example 12.1. Finding the Center of Gravity

st^1 pivot

nd^2 pivot

Explain how you can find an object’s center of gravity by suspending it from a string.

GOT IT? 12.2.

The dancer in the figure is balanced; that is, she’s in static equilibrium.Which of the three lettered points could be her center of gravity?

Example 12.2. Ladder Safety

A ladder of mass

m^ & length

L^ leans against a frictionless wall.

The coefficient of static friction between ladder & floor is

.

Find the minimum angle

^ at which the ladder can lean without slipping.

Fnet x

:^

1

2

n^

n

^

^

Fnet y

:^

1

n^

m g 

Choose pivot point at bottom of ladder. :^ z^

^

^

^



sin 180 2

sin 90

L 2

L n

m g

 ^

^

 ^

2

1 n^

n 

m g 



sin 2

cos

L 2

L n

m g

^

2

tan^

m g^2 n

^ 

0



^ 

90 

y

x



 m g

n^1

f =S^

 n^1 n^2 i

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Example 12.3. Arm Holding Pumpkin

Find the magnitudes of the biceps tension & the contact force at the elbow joint.

Fnet x

:^

cos

F^ c x

T

^

Fnet y

:^

sin^

c y

T^

F^

m g

M g

^ 

^

^

Pivot point at elbow. :^ z^

1

2

3

sin^

x T

x m g

x M g

^ 

^

^

^



2

3 sin 1 x^ m

x M

g

T^

x

^

^ ^

^

^ ^



^

2

2.^

4.^ 9.^

/

sin 80

m^

kg^

m^

kg^

m^ s

m 

^



^



^



N

cos F^ c x

T

^



sin F^ c y

T^

m^

M^

g

^

^

 500

cos N ^

^

N

^

^

^

 ^

2 

500

sin 80

2.^ 4.^ 9.^

/

N^

kg^

kg^

m^ s

 ^

 ^

^

N

^ 

2

2

c^

c x^

c y

F^

F^

F

^

^

N

^

~ 10

M g

y

x M g

m g

T

F c

80 

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Application: Statue of Liberty

Sculptor Bartholdi : lasting as long as the pyramids.Deviation from Eiffel’s plan resulted in excessive torque.Major renovation was required after only 100 yrs.

12.4. Stability

Stable equilibrium:

Original configuration regained after small disturbance.

Unstable equilibrium: Original configuration lost after small disturbance.

Stable equilibrium

unstable

equilibrium

Stable equilibrium :PE at global min

Metastable equilibrium :PE at local min

Example 12.4. Semiconductor Engineering

A new semiconductor device has electron in a potential

U (

x ) =

a x

2 –^

(^4) b x ,

where

x^ is in nm,

U^

in aJ (

^18 J),

a^ = 8 aJ / nm

2 ,^ b

= 1 aJ / nm

Find the equilibrium positions for the electron and describe their stability.

Equilibrium criterion :

d Ud x



3

2

a x

b x 

2 nm ^ 

x^ 

a 2 x^

b   or

^

aJ^

nm aJ^

nm

2

2

2

d^ U

a^

b x

d x

2 2

(^20)

0 x d^ U

a

d x^

 

^

x = 0 is (meta) stable

2 2

/

4

0

x^

a^ b d^ U

a

d x^

 

 ^

^

x =^ 

(a/2b) are unstable

equilibria Metastable

GOT IT? 12.

Which of the labeled points are stable, metastable, unstable, or neutrally stable equilibria?

S

M

U

U

N