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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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Monday,September 17, 2012 Moments in 2D 3
Tools
Basic Trigonometry Pythagorean Theorem Algebra Visualiza-on Posi-on Vectors Unit Vectors
Monday,September 17, 2012 Moments in 2D 4
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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Monday,September 17, 2012 Moments in 2D 5
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)
Monday,September 17, 2012 Moments in 2D 6
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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Monday,September 17, 2012 Moments in 2D 7
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
Monday,September 17, 2012 Moments in 2D 8
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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Monday,September 17, 2012 Moments in 2D 9
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=
Monday,September 17, 2012 Moments in 2D 10
Defini-on
A two dimensional example Take the moment of F about a
M d F=
Fax
y
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Monday,September 17, 2012 Moments in 2D 11
Defini-on
First we develop the line of ac-on of F
M d F=
Fax
y
Monday,September 17, 2012 Moments in 2D 12
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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Monday,September 17, 2012 Moments in 2D 13
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
Monday,September 17, 2012 Moments in 2D 14
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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Monday,September 17, 2012 Moments in 2D 15
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
Monday,September 17, 2012 Moments in 2D 16
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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Monday,September 17, 2012 Moments in 2D 17
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=
Monday,September 17, 2012 Moments in 2D 18
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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Monday,September 17, 2012 Moments in 2D 19
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
Monday,September 17, 2012 Moments in 2D 20
Defini-on
Looking at the triangle formed we can state
M d F=
Fa x
y
dperpendicular
d
( )sind d =
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Monday,September 17, 2012 Moments in 2D 21
Defini-on
So another way to calculate the magnitude is
M d F=
Fa x
y
dperpendicular
d
( )sinM d F=
Monday,September 17, 2012 Moments in 2D 22
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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Monday,September 17, 2012 Moments in 2D 23
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
Monday,September 17, 2012 Moments in 2D 24
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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Monday,September 17, 2012 Moments in 2D 25
Defini-on
CWCCW
Monday,September 17, 2012 Moments in 2D 26
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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Monday,September 17, 2012 Moments in 2D 27
Defini-on
Draw the clock face so that the dperpendicular is the minute hand of the clock
CWCCW
Fa x
y
dperpendicular
d
Monday,September 17, 2012 Moments in 2D 28
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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Monday,September 17, 2012 Moments in 2D 29
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
Monday,September 17, 2012 Moments in 2D 30
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.
Monday,September 17, 2012 Moments in 2D 31
Determine the moment of the force about point O. Neglect the thickness of the member.
Problem F4-6.
Monday,September 17, 2012 Moments in 2D 32
Determine the moment of the force about point O.
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Problem F4-7.
Monday,September 17, 2012 Moments in 2D 33
Determine the resultant moment produced by the forces about point O.
Problem F4-4.
Monday,September 17, 2012 Moments in 2D 34
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)
4-15
Monday,September 17, 2012 Moments in 2D 35
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