Physics 106 Lesson #2

Static Equilibrium

Dr. Andrew Tomasch

2405 Randall Lab

atomasch@umich.edu

• Scalars: magnitude only (ex: radius R =10 cm)• Vectors: magnitude and direction

– Arrow length ≡ magnitude of the vector

– Arrow direction ≡ vector direction

Example:

Displacement ≡

= 1 m to the right

d

d

magnitude direction

Last Time: Scalars & Vectors

1 md

The Net (Total) Force

• Because the washer is at rest the net (total) force on it is zero

• The net force is the vector sum of all the forces acting on the object

• How do we add vectors?

Washer

WeightW = mg

Spring ForceFs= kx

?

• Place vectors “tip to tail” and draw an

arrow from the start (tail) of one vector to the end (tip) of the other

Vector Addition

?A B

A

BA+B

• How about A-B ?

A

B-B A-B

A-B = A+(-B)

CautionQuiz

Ahead

Concept Test #1

Two vectors, one with magnitude 3 m and the other with magnitude 4m, are added together. The resultant vector could have a magnitude as small as:

A) 1m

B) 3m

C) 5m

D) 7m

The resultant has the largest magnitude when the two vectors are parallel and the smallest when they are antiparallel.

4 m

3 m

1 m

CautionQuiz

Ahead

Concept Test #2

Two vectors have unequal magnitudes. Can their sum be zero?

• Yes

• No The smallest possible vector sumIs when the two vectors are antiparallel.The only way for this sum to be zero isIf the vectors have equal magnitudes.

0

• For a body to be in equilibrium and remain at rest:

• An example: 0F

Equilibrium Condition for a Body at Rest

0F

Washer

WeightW = mg

Spring ForceFs= kx

+ =0Shorthand forthe net force:“The vector sumof all forces”

• Torque (a vector): Magnitude ≡ the product of a perpendicular force and the distance r to a rotation axis:

rF

rF is perpendicular to r

Torque:

Torque can also be defined more generally for forces that are not perpendicular to r by expressing F as the sum of two component vectors, one parallel to the door and one perpendicular to it. Only the perpendicular component vector produces a torque about the hinge to make the door rotate.

"The magnitude

(length) of vector "

A force which points through the rotationaxis produces no torque because r = 0!

The distance r is also called the lever arm

F

F

F

F r

Torque Continued• Direction (looking down the rotation axis):

– Counterclockwise rotation is “(+)” (positive)– Clockwise rotation is “(-)” (negative)– Direction in space by Right Hand Rule (RHR)

• Why a perpendicular force F ? – Because only a perpendicular force can produce a

rotation

• Units:

rF

mN points

out of page

CautionQuiz

Ahead

FArrowout of page

Arrowinto page

Concept Test #3

You are trying to open a door that is stuck by pulling on the knob in a direction perpendicular to the door. If you instead tie a rope to the knob and pull with the same magnitude force in the same direction, does the torque you exert increase?

• Yes• No

r

Both r and F remain unchanged. F does not change magnitude or direction → torque remains unchanged.

F

Concept Test #4You are using a wrench to try to tighten a nut. Which of the arrangements shown is most effective for tightening the nut? List in order of decreasing effectiveness. (The “rod” can be used to extend the wrench handle length). The force F is the same in each case.

F

1

F

rod

2

F

3

F

rod

4

A. 2 > 1 > 3 > 4

B. 2 > 1 = 4 > 3

C. 4 > 2 > 1 > 3

D. 2 > 3 > 4 > 1

The longest lever arm rexerts the largest torque

for the same applied force

r

rr r

Equilibrium for Rigid Bodies• Extended objects

which do not change shape are called rigid bodies

• For a rigid body to be in equilibrium:

• For a rigid body to remain stationary in one place, the net force acting on it must be zero

• For a rigid body to remain stationary and not rotate the net torque about any axis through the body must be zero

• The net force/torque is the vector sum of all forces/torques acting on the body (analyze with Free Body Diagram)

0

0

F

Center of Mass/Gravity• There is a special geometrical point associated with

any rigid body (extended object) where we can attribute the body’s mass and weight as acting at that point

• This special point is called the center of mass (CM) and the center of gravity (CG) of the object

• The definitions for center of mass and center of gravity are different, but if the acceleration of gravity g is constant over the extent of the object, then the center of mass and the center of gravity are located at the same geometrical point. For uniform objects this will be the geometrical center of the object

• The motion of any rigid body of mass m can be described as the trajectory through space of a point mass m located at its center of mass on which is superposed rotation of the object about the center of mass

• For the purpose of calculating torques, all the weight of the extended object can be assumed to act at the center of gravity Demo: Dumbbell With Lights

It is at the center of the circular ring, half way from thebottom of the donut - where there is no dough!

Center of Gravity (CG)

Where Is the Center of Gravity of This Yummy

Donut?

The Center of gravity does not need to be located inside the object. Sometimes the CG can be located in a surprising place…

After A Bite…

Center of Gravity

If it falls within the object, the center of gravitycorresponds to the point where an object balances.

• Treat the wine rack and the bottle as a rigid body

• Two forces:– Supporting force FN from

table– Weight at the center of

gravity of the system• Net torque = 0→No rotation →• The center of gravity is

located exactly above the point where the supporting force acts

• The supporting force FN from the table is also called the “Normal” (perpendicular) force

Wine Bottle Balance

FN

Axis of rotation

W

CG

Neither force produces a torqueabout the Axis of Rotation sinceboth forces point through the axisso that r = 0 for both forces

• The wine bottle and seesaw are balanced in the same way!

Wine Bottle Balance

FN

Axis of rotation

W

CG

Axis of rotation W

FN

CG