3rdLE Lecture 31 - R11 Static Equilibrium (4)

7/25/2019 3rdLE Lecture 31 - R11 Static Equilibrium (4)

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Lecture 30: Center of gravity and static equilibrium

Objectives1. Compare and contrast the center of mass, center of gravity,and stability.

2. Enumerate the necessary and sufficient conditions for staticequilibrium.3. Solve static equilibrium problems, such as see-saw, mobile,cable-hinge-strut system, leaning ladder, and wheel climbing a

step.


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For everyday objects, the center of gravity is the same as the center of mass.

Recall: Center of mass

Center of gravity, the average position of all the particles ofweight that make up an object.

The center of mass of an object is the point located

at the objects average position of mass .


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There can be a small differencebetween center of gravity andcenter of mass when an object islarge enough for gravity to varyfrom one part to another

Example: The center of gravity of theSears Tower in Chicago is about 1 mmbelow its center of mass because thelower stories are pulled a little morestrongly by Earths gravity than the upperstories.


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4

Center of gravity (CG) is the center of mass (CM)IF acceleration due to gravity is constant throughout

Center of Gravity: Point where allgravitational force seems to act

Location where weight is concentratedWhen body is supported at this point,

no net torque due to gravity

Finding CG:- Suspend object at least two

(2) different points- Draw vertical lines madewhen supported- The intersection of lines: CG

CG


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The blocktopples when

the CG extendsbeyond its

support base.

CG and stability: Rule for TopplingIf the CG extends outside the area of support, an unbalanced

torque exists, and the object will topple.

You can lean overand touch your

toes withouttoppling only ifyour CG is abovethe area boundedby your feet.


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Recall: A particle in equilibrium : it does NOT

accelerate ( = )

Conditions for Static Equilibrium

Torque: tendency to rotate

No rotation: = An object is static equilibrium does not accelerate and does not rotate


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Example 1:


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Example 2:


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Example 3:


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Solving equilibrium problemsStatic equilibrium:

=

=

Torques thatproduce

clockwise rotation

Torques that producecounter clockwise

rotation=

two equationsmean the same


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Sample Problem 1: A pair of torques can balance eachother. Balance is achieved if the torque that tends to produce

clockwise rotation by the boy equals the torque that tends toproduce counterclockwise rotation by the girl. What must bethe weight of the boy to achieve static equilibrium?

Equilibrium: Balanced Torques

Torque dueto girl isCCW:

(positive)

Torque dueto boy is CW:(negative)


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Torque due

to girl isCCW:(positive)

Torque due

to boy is CW:(negative)

= 0girl + boy + seesaw = 0600Nm wb(1.5m)= 0

wb = 400N

GIRL: CCWtorque = RgFg(sin90) = (200N)(3m) = 600NmBOY: CWtorque = RbFb(sin90)=(w boy )(1.5m)SEESAW: 0 because R = 0 (at the center of mass)


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Sample Problem 2: Weight distribution for the carAn auto magazine reports that a certain sports car has 53% of itsweight on the front wheels and 47% on its ear wheels, with a

2.46-m wheelbase. This means that the total normal force on thefront wheel is 0.53w and that on the rear wheel is 0.47w, wherew is the total weight. The wheelbase is the distance between thefront and the rear axles. How far on front of the rear axle is the

cars center of gravity?


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No x-component of forceSince 0.47w + 0.53w + (-w) = 0

1st condition for equilibrium satisfied, but it does not contain L cg

FBD of the car

2nd condition for equilibrium: = 0 (rear wheel : pt. of rotation)

Lcg is between the 2 supports


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Sample Problem 3:


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Apply the first and second conditions of equilibrium to Clea

= 0

0.95 = 0

= . ,( .

=89 =157 + =

=

The normal force at her feet is greater than at her rear

feet, so her center of gravity is closer to her front feet

= 0


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VIDEO DEMONSTRATION: BALANCING ACT USING STICKS & A FEATHER

This video performance demonstrates the concepts of center of mass,torque, and balance in a dramatic fashion. The performer a series ofsticks of increasing size, starting with a feather and ending with a stickabout 2 m in height. The precise nature of this feat is demonstrated atthe end of the video.


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How he did it ??? (thinking in Mandarin)

He carefully placed the stick in the center of mass of the feather;

then placed another stick in the the center of mass of the feather - stick body;then placed another stick in the feather stick stick body ; and so on

= ......


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How he did it??? (ENGLISH TRANSLATION)

He carefully placed the stick in the center of mass of the feather;

then placed another stick in the the center of mass of the feather - stick body;then placed another stick in the feather stick stick body ; and so on

= so torques must be equal toachieve static equilibrium... now, where is

the center of mass??


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=

Summary: Center of Mass & Static Equilibrium

= Conditions for static equilibrium:

Center of Gravity:point where allgravitational forceseems to act


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=132.40 10 = 0.80 10

= A (0.80x10 8Pa) = (3.0x10 -4 m 2)(0.80x10 8Pa) = 2.40x10

4N

F = ma gives - mg = ma= =2.40 101200 9.8 /2= . .

= A = 0.50cm

2

= 0.50 x 10-4

m2

= =(4.00 )(5000 )

(0.50 10 2)(0.20 10 )= .

Homework: SW # 3

Seatwork 4


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RECIT 11 (TAKE HOME, by pair/individual)Due: 23 October

Problem Set Ch 6,7,8 will be emailed thisweekend

Due: 27 October

2nd Long exam: 26 October (M)12:00PM to 02:00PM(R)11:00PM to 01:00PM(E) see me after class01:00PM to 03:00PM(L) see me after class


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Seatwork- solve problems in yournotebooks- write the answers only inyour bluebook- indicate the date

January 6, 20141. Blah?

2. Blah blah!

3. Blah blah blah!

4. Blah blah blah blah!

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HOMEWORK:


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HOMEWORK:1. A box with its center of mass off-center as indicated bythe dot is placed on an inclined plane. In which of thethree orientations shown, if any, does the box tip over?

2. Which of the following situations satisfy the conditions for staticequilibriuma. A stationary rod pivoted at one end.b. A bowling ball rolls without slipping at a constant non-zerospeed along a horizontal alleyc. A cable wrapped around a cylinder that can rotate about its axisunwinds without slipping at a constant speed


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A B

3: Draw the FBD of the situation

4: Calculate the maximum weight you can place withoutbreaking the cable. (use 1 st condition)5: Where should you put the weight? (use 2 nd condition, calculate for CG)

Weight, w

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3rdLE Lecture 31 - R11 Static Equilibrium (4)