Moments in 2D - Memphis - University of Memphis in 2D.pdf · moment MdF= ⊥ 18 Moments in 2D...

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He who asks is a fool for five minutes, but he who does not ask remains a fool forever.

-Chinese proverb

Moments

Monday,September 17, 2012 Moments in 2D 2

Objec-ves

Understand what a moment represents in mechanics

Understand the scalar formula-on of a moment

Understand the vector formula-on of a moment

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Tools

Basic Trigonometry Pythagorean Theorem Algebra Visualiza-on Posi-on Vectors Unit Vectors


Defini-on

A moment is the tendency of a force to cause rota-on about a point or around an axis

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Defini-on

When we discussed forces earlier, we looked at their tendency to cause transla'on (movement along an axis)

Now we are looking at their tendency to cause rota'on (movement around an axis)


Defini-on

Moment is oKen used in the same sense as torque which is also the tendency to rotate.

We will use moment exclusively in this class

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Defini-on

Moment is dependent on both the magnitude of the force and how far away the force is from the point or axis the rota-on is occurring about


Defini-on

The magnitude of the moment is the product of the perpendicular distance to the line of ac-on of the force from the point or axis around which the rota-on is taking place and the magnitude of the force

M d F=

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Defini-on

No-ce lMagnitude of the moment lPerpendicular distance from the point or axis about which rota-on is taking place to the line of ac-on of the force

lMagnitude of the force

M d F=


Defini-on

A two dimensional example Take the moment of F about a

M d F=

Fax

y

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Defini-on

First we develop the line of ac-on of F

M d F=

Fax

y


Defini-on

Then we can draw a perpendicular line from a to the line of ac-on of F

M d F=

Fa

x

y

dperpendicular

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Defini-on

And use the length of that line and the magnitude of the force to calculate the magnitude of the moment

M d F=

Fa

x

y

dperpendicular


Defini-on

No-ce that the magnitude of the moment is the scalar product of a distance and a force

M d F=

Fa

x

y

dperpendicular

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Defini-on

That makes the units of magnitude for a moment l Ft-lbs lN-m

The order of terms doesnt maPer

M d F=F

ax

y

dperpendicular


Defini-on

The point about which rota-on would occur is known as the moment center

In this example, a is the moment center

M d F=F

ax

y

dperpendicular

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Defini-on

If we dont know the perpendicular distance but we can construct some other distance from the moment center to the line of ac-on of the force, we can s-ll calculate the moment

M d F=


Defini-on

We can construct some distance d from the moment center to the line of ac-on of the force

M d F=

Fa x

y

dperpendicular

d

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Defini-on

Find the angle, , that the new moment arm, d, makes with the line of ac-on of the force

M d F=

Fa x

y

dperpendicular

d


Defini-on

Looking at the triangle formed we can state

M d F=

Fa x

y

dperpendicular

d

( )sind d =

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Defini-on

So another way to calculate the magnitude is

M d F=

Fa x

y

dperpendicular

d

( )sinM d F=


Defini-on

The direc-on of the moment can be described in a two-dimensional problem as either clockwise CW, or counter-clockwise CCW

By conven-on, we label CW moments as nega-ve and CCW moments as posi-ve

You will see why when we do three-dimensional problems

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Defini-on

One way to see the sense of rota-on is to think of a clock face on an old clock

The large arm is the minute hand, the smaller one is the hour hand


Defini-on

If something pushes the minute hand where -me passes correctly, then it is moving the hand clockwise CW

If something pushes the minute hand where -me passes backwards, then it is moving the hand counter-clockwise CCW

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Defini-on

CWCCW


Defini-on

Now we can use the clock to determine the sense of rota-on of the moment

We start by placing the center of the clock on the moment center

CWCCW

Fa x

y

dperpendicular

d

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Defini-on

Draw the clock face so that the dperpendicular is the minute hand of the clock

CWCCW

Fa x

y

dperpendicular

d


Defini-on

Determine that if F were pulling or pushing on the minute hand would -me be passing normally or backwards

CWCCW

Fa x

y

dperpendicular

d

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Defini-on

In this case F would be causing -me to pass backwards so the moment is CCW and therefore a posi-ve moment

CWCCW

Fa x

y

dperpendicular

d


Resultant Moment

If more than one moment is ac-ng about a point, then the resultant moment is the sum of the individual moments

The sign of the resultant is develop by the signs of the individual moments using the conven-on we developed earlier.

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Problem F4-4.


Determine the moment of the force about point O. Neglect the thickness of the member.

Problem F4-6.


Determine the moment of the force about point O.

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Problem F4-7.


Determine the resultant moment produced by the forces about point O.

Problem F4-4.


Determine the moment of the force about point O.

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Homework

Problem 4-7 Problem 4-10 (The resultant moment is the sum of the moments)

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Moments in 2D - Memphis - University of Memphis in 2D.pdf · moment MdF= ⊥ 18 Moments in 2D...

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Transcript of Moments in 2D - Memphis - University of Memphis in 2D.pdf · moment MdF= ⊥ 18 Moments in 2D...