Moments “He who asks is a fool for five minutes, but he who does not ask remains a fool forever.” -­‐Chinese proverb 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 2
Moments in 2D
Monday,September 17, 2012
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Tools ¢  Basic Trigonometry ¢  Pythagorean Theorem ¢  Algebra ¢  Visualiza-on ¢  Posi-on Vectors ¢  Unit Vectors 3
Moments in 2D
Monday,September 17, 2012
Defini-on ¢  A moment is the tendency of a force to cause rota-on about a point or around an axis 4
Moments in 2D
Monday,September 17, 2012
2
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) 5
Moments in 2D
Monday,September 17, 2012
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 6
Moments in 2D
Monday,September 17, 2012
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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 7
Moments in 2D
Monday,September 17, 2012
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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Moments in 2D
Monday,September 17, 2012
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Defini-on ¢  No-ce l  Magnitude of the moment l  Perpendicular distance from the point or axis about which rota-on is taking place to the line of ac-on of the force l  Magnitude of the force M = d⊥ F
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Moments in 2D
Monday,September 17, 2012
Defini-on ¢  A two dimensional example ¢  Take the moment of F about a y
F
a
M = d⊥ F
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Moments in 2D
x
Monday,September 17, 2012
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Defini-on ¢  First we develop the line of ac-on of F y
F
a
M = d⊥ F
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Moments in 2D
x
Monday,September 17, 2012
Defini-on ¢  Then we can draw a perpendicular line from a to the line of ac-on of F y
F
dperpendicular
a
M = d⊥ F
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Moments in 2D
x
Monday,September 17, 2012
6
Defini-on ¢  And use the length of that line and the magnitude of the force to calculate the magnitude of the moment y
dperpendicular
F
a
M = d⊥ F
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Moments in 2D
x
Monday,September 17, 2012
Defini-on ¢  No-ce that the magnitude of the moment is the scalar product of a distance and a force y
F
dperpendicular
a
M = d⊥ F
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Moments in 2D
x
Monday,September 17, 2012
7
Defini-on ¢  That makes the units of magnitude for a moment l  Ft-­‐lbs l  N-­‐m ¢  The order of terms doesn’t maPer y
dperpendicular
F
M = d⊥ F
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a
Moments in 2D
x
Monday,September 17, 2012
Defini-on ¢  The point about which rota-on would occur is known as the moment center ¢  In this example, a is the moment center y
F
M = d⊥ F
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Moments in 2D
dperpendicular
a
x
Monday,September 17, 2012
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Defini-on ¢  If we don’t 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
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Moments in 2D
Monday,September 17, 2012
Defini-on ¢  We can construct some distance d from the moment center to the line of ac-on of the force y
F
M = d⊥ F
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Moments in 2D
dperpendicular
θ
d
a x
Monday,September 17, 2012
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Defini-on ¢  Find the angle, θ, that the new moment arm, d, makes with the line of ac-on of the force y
F
dperpendicular
θ
d
M = d⊥ F
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Moments in 2D
a x
Monday,September 17, 2012
Defini-on ¢  Looking at the triangle formed we can state d ⊥ = d sin (θ )
F
M = d⊥ F
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Moments in 2D
y
dperpendicular
θ
d
a x
Monday,September 17, 2012
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Defini-on ¢  So another way to calculate the magnitude is M = d sin (θ ) F
F
M = d⊥ F
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y
dperpendicular
θ
d
Moments in 2D
a x
Monday,September 17, 2012
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 22
Moments in 2D
Monday,September 17, 2012
11
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 23
Moments in 2D
Monday,September 17, 2012
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 24
Moments in 2D
Monday,September 17, 2012
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Defini-on CCW
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CW
Moments in 2D
Monday,September 17, 2012
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 y
F
dperpendicular
θ
CCW
d
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Moments in 2D
a x
CW
Monday,September 17, 2012
13
Defini-on ¢  Draw the clock face so that the dperpendicular is the minute hand of the clock y
dperpendicular
F
θ
CW
d
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CCW
a x
Moments in 2D
Monday,September 17, 2012
Defini-on ¢  Determine that if F were pulling or pushing on the minute hand would -me be passing normally or backwards y
F
dperpendicular
θ
CW
d
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Moments in 2D
CCW
a x
Monday,September 17, 2012
14
Defini-on ¢  In this case F would be causing -me to pass backwards so the moment is CCW and therefore a posi-ve moment y
F
dperpendicular
θ
CW
d
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Moments in 2D
CCW
a x
Monday,September 17, 2012
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. 30
Moments in 2D
Monday,September 17, 2012
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Problem F4-­‐4. Determine the moment of the
force about point O. Neglect the
thickness of the member.
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Moments in 2D
Monday,September 17, 2012
Problem F4-­‐6. Determine the moment of the
force about point O.
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Moments in 2D
Monday,September 17, 2012
16
Problem F4-­‐7. Determine the resultant moment
produced by the forces about
point O.
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Moments in 2D
Monday,September 17, 2012
Problem F4-­‐4. Determine the moment of the
force about point O.
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Moments in 2D
Monday,September 17, 2012
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Homework ¢  Problem 4-­‐7 ¢  Problem 4-­‐10 (The resultant moment is the sum of the moments) ¢  4-­‐15 35
Moments in 2D
Monday,September 17, 2012
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