[email protected] engr-36_lec-07_moments_intro.ppt 1 bruce mayer, pe engineering-36:...

[email protected] • ENGR-36_Lec-07_Moments_Intro.ppt1

Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 36

Chp 4: Intro to

Moments



Moment (Torque) Described

In Physics and Engineering a MOMENT is a measure of TWISTING Power

The MAGNITUDE of a Moment is the PRODUCT of a Lever Arm Distance and an Intensity• The “Intensity” can be a Force, an Electric

Charge, an Area, a Mass, or other• In Engineering Mechanics

the Intensity takes the form of a Force

http://www.answers.com/main/Record2?a=NR&url=http://commons.wikimedia.org/wiki/Image:Fysik%20vridmoment.png



Moment Described

In General, MOMENTS are VECTOR Quantities with Magnitude (see previous slide) and Direction

The Direction of the Moment Vector is determined by the Right Hand Rule• Wrap Fingers in the Direction of

ROTATION (or tendency to rotate), then THUMB points in the Direction of the Moment Vector



Moment Center

The MOMENT CENTER is equivalent to the PIVOT POINT about which Rotation would occur upon application of a Force whose Line of Action is OFFSET from the Pivot Point

Moment Center (MC)or Pivot Point

distance from the Pivot to Force Loa



Position Vector

The Position Vectorruns from theMoment Center toANY Point on theLoA of the Force• Often times the most Convenient Point on

the LoA is the Point of Application (PoA)– i.e., Pos. Vector runs from the Pivot to the PoA

The position Vector Contains within it the “LeverArm” part of the Moment Calc



Picking the Position Vector, r

Often the Most Convenient Position Vector is that which runs From the Pivot to the Point of Application

r runs from the Pivot to the Point

OTHER Pts on the Force LoA may be more easily determined and are thus More Convenient



Shortest Position Vector

The Shortest vector is that which is to the Force LoA

The Mag of the Shortest r is called the Perpendicular Distance, d:

distance yields Shortest r

minrd



Moment Magnitude

As Noted Previously the Magnitude of a Moment is related to the product of• The Position Vector, r• The Force, F

r

Mathematically

FdM FsinrsinFrM

• Thus knowing F & d allows Calc of the Moment magnitude, but NOT its SENSE (direction)



Moment Sense

TWO-DIMENSIONAL STRUCTURES Have Length And Breadth But Negligible Depth And Are Subjected To Forces Contained In The PLANE Of The Structure The Plane Of The Structure Contains The Point O And The Force F. MO, The Moment Of The Force About O Is Perpendicular To The Plane.



Moment Sense/Direction

If The Force Tends To Rotate The Structure CLOCKWISE, The Sense Of The Moment Vector Is INTO The Structure Plane• SIGN{MO} → NEGATIVE

If The Force Tends To Rotate The Structure COUNTER-clockwise, The Sense Of The Moment Vector Is OUT Of The Structure Plane • SIGN{MO} → POSITIVE



Moment Direction by Rt Hand Rule

1. Point Fingers in r Direction

2. Curl Fingers Toward +F Direction

3. THUMB Points in the Direction of M

k̂

k̂

HINT:Put r & FTail-to-Tail

rF



Moments Point in ALL Directions Since r and F

can be arbitrarily oriented relative to the CoOrd Axes, then M will also be arbitrarily Oriented

• Confirm These using your own Right Hand



Moment Units Force • Dist

Discern the UNITS for Moments from

FdM FsinrsinFrM

ForceDistanceMUnits

Typical Units• Ft-lbs• In-lbs• N-m• N-mm



Example: Moment Calculation A 100-lb Vertical Force Is Applied To

The End Of A Lever Which Is Attached To a Shaft At O. DETERMINE

a) Moment About O

b) Horizontal Force At Pt-A Which Creates The SAME Moment

c) Smallest Force At Pt-A Which Produces The SAME Moment

d) Location For a 240-lb Vertical Force To Produce The SAME Moment

e) Whether Any Of The Forces From b, c, and d is EQUIVALENT To The ORIGINAL Force



Example M Calc – Soln (a) Moment About O Is Equal To The Product

Of The Force And The PERPENDICULAR DISTANCE Between The Line Of Action Of The Force And O

The Force Tends To Rotate The Lever CLOCKWISE, Thus The Moment Vector points INTO The Plane Of The Paper• The Moment Vector Qty is thus NEGATIVE

in. 12lb 100

in. 1260cosin.24

O

OO

M

d

FdMMin lb 1200 OM



Example M Calc – Soln (b) Horizontal Force at A That Produces

The Same Moment

in. 8.20

in. lb 1200

in. 8.20in. lb 1200

in. 8.2060sinin. 24

F

F

FdM

d

O

lb 7.57F

60



Example M Calc – Soln (c) The Smallest Force at A To Produce

The Same Moment Occurs When The Perpendicular Distance is a Maximum • i.e., When F Is Perpendicular To OA

in. 42

in. lb 1200

in. 42in. lb 1200

F

F

FdMO

lb 50F



Example M Calc – Soln (d) To Determine The Point Of Application

Of A 240 lb Vertical Force To Produce The Same Moment

in. 5cos60

in. 5lb 402

in. lb 1200

lb 240in. lb 1200

dOB

d

d

FdMO

in. 10OB

cos60in. 5or OB



Example M Calc – Soln (e) Although Each Of The Forces In Parts b),

c), and d) Produces The Same Moment As The 100 lb Force, NONE Are of The Same MAGNITUDE And SENSE (Line of Action) as the original pull

\ NONE Of The Forces Is Equivalent To The 100 lb force



WhiteBoard Work

Let’s WorkProblem

4.21

In order to pull out the nail at B, the force F exerted on the handle of the hammer must produce a clockwise moment of 500 in∙lb. about point A. Determine the required magnitude of force F.



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 36

Appendix

[email protected] engr-36_lec-07_moments_intro.ppt 1 bruce mayer, pe engineering-36:...

Documents

pe engineering

moment vector slide

moment calc slide

force slide

shortest vector

force loa slide

convenient position

pivot point moment center