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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]

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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

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

Moment Center

distance from the Pivot to Force Loa

- 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

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

Often the Most Convenient Position Vector is that which runs From the Pivot to the Point of ApplicationPicking the Position Vector, r

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

r runs from the Pivot to the Point

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

As Noted Previously the Magnitude of a Moment is related to the product of

The Position Vector, r

The Force, F

Moment Magnitude- Mathematically

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

TWO-DIMENSIONAL STRUCTURES Have Length And Breadth But Negligible Depth And Are Subjected To Forces Contained In The PLANE Of The StructureMoment Sense

- 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 COUNTER-clockwise, The Sense Of The Moment Vector Is OUT Of The Structure Plane
- SIGN{MO} → POSITIVE

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

Moment Direction by Rt Hand Rule

- Point Fingers in r Direction
- Curl Fingers Toward +F Direction
- THUMB Points in the Direction of M

HINT:

Put r & FTail-to-Tail

F

r

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

A 100-lb Vertical Force Is Applied To The End Of A Lever Which Is Attached To a Shaft At O. DETERMINE

Moment About O

Horizontal Force At Pt-A Which Creates The SAME Moment

Smallest Force At Pt-A Which Produces The SAME Moment

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

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

Example: Moment CalculationMoment 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

Example M Calc – Soln (a)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

Example M Calc – Soln (c)To Determine The Point Of Application Of A 240 lb Vertical Force To Produce The Same MomentExample M Calc – Soln (d)

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 pullExample M Calc – Soln (e)

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

WhiteBoard Work

Let’s WorkProblem4.21

In order to pull out the nail at B, the force Fexerted 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.

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