section 4.1 moment of a force (page 121)mattison/courses/phys170/p170-n4.pdf · section 4.1 moment...

Section 4.1 Moment of a Force (page 121)

MOMENT

!M

O OF A FORCE

!F (page 1)


MOMENT

!M

O OF A FORCE

!F (page 2)

A force

!F is applied to a body. O is a fixed point.

Torque or moment

!M

O is the tendency of

!F to cause rotation of

the body about O.

• Magnitude Law of the Lever

The magnitude of

!M

O is the product of the magnitude of

!F and

the lever arm d:

MO= Fd

The lever arm d is the perpendicular distance between O and the

line of action of !F .

!F can only cause rotation when d ! 0 .

Archimedes discovered the Law of the Lever in 240 BC.

• Direction Axis of Rotation

The direction of

!M

Ois the axis of rotation.

!M

O is perpendicular to the plane containing

!F and d.

The direction of

!M

O is given by the right-hand rule.

See Figure 4-2 (a).


MOMENT

!M

O OF A FORCE

!F (page 3)

• Definition

!M

O=!r !!F


MOMENT

!M

O OF A FORCE

!F (page 4)

• Definition

!M

O=!r !!F

The magnitude and direction of

!M

O follow from the definition of

the cross product of two vectors.

Magnitude MO= r F sin!

r is the magnitude of !r F is the magnitude of

!F

! is the angle between the tails of !r and

!F (0 !" !180

o

).

d = rsin! is the lever arm for !F .

Direction

!M

O!!r

!M

O!!F

!M

O points along the axis of the rotation caused by

!F .

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MOMENT

!M

P OF A FORCE

!F (page 5)

CARTESIAN VECTOR

• A force

!F = F

x

!i + F

y

!j + F

z

!k acts at point A( x

A, y

A, z

A).

• The moment

!M

P of F!"

about point P( xP, y

P, z

P) is

!M

P=!r !!F

where

!r =!rA!!rP= (x

A! x

P)!i + (y

A! y

P)!j + (z

A! z

P)!k

• Written as a Cartesian vector,

!M

P= M

x

!i + M

y

!j + M

z

!k

where

Mx= (y

A! y

P)F

z! (z

A! z

P)F

y

My= (z

A! z

P)F

x! (x

A! x

P)F

z

Mz= (x

A! x

P)F

y! (y

A! y

P)F

x

• Alternatively,

!M

P=!r !!F =

!i

!j

!k

xA"xP yA"yP zA"zP

Fx Fy Fz

Section 4.2 Cross Product

CROSS PRODUCT OF TWO VECTORS (page 121) page 1

CROSS PRODUCT OF TWO VECTORS (page 121) page 2

!C =!A !!B

• Magnitude C = ABsin!

A is the magnitude of

!A B is the magnitude of

!B

! is the angle between the tails of

!A and

!B (0 !" !180

o

)

• Direction

!C !

!A

!C !

!B

Tail-to-Tail rule

Push

!A to

!B with your fingers of your right hand.

Your thumb points along the direction of !C .

• Cross product of Cartesian unit vectors

!i !!i = 0

!i !!j =!k

!j !!i = "

!k

• Cross product of Cartesian vectors

!A = A

x

!i + A

y

!j + A

z

!k

!B = B

x

!i + B

y

!j + B

z

!k

!C =!A !!B =

!i

!j!k

Ax Ay Az

Bx By Bz

Section 4.5 Moment of a Force About a Specified Axis (page 139)

MOMENT OF A FORCE ABOUT A SPECIFIED AXIS (page 1)

• Figure 4-21 shows the projection

!M

a along the axis a of the moment

!M

O of the force

!F . O( x

O, y

O, z

O) is any point on the axis a.

A( xA, y

A, z

A) is any point on the line of action of

!F .

• Section 4.5 Moment of a Force About a Specified Axis (page 139)

MOMENT OF A FORCE ABOUT A SPECIFIED AXIS (page 2)

• Definition:

!M

a= M

a

!ua

M

a=!uai

!M

O=!uai (!r !!F)

!ua is a unit vector along the axis a

• Cartesian vector form:

!ua= u

ax

!i + u

ay

!j + u

az

!k

!r =!rA!!rO= (x

A! x

O)!i + (y

A! y

O)!j + (z

A! z

O)!k

!F = F

x

!i + F

y

!j + F

z

!k

Ma=

uaxuay

uaz

xA!xO yA!yO zA!zO

Fx Fy Fz

Section 4.6 Couple Moment

COUPLE

• Couple

A couple is defined as two anti-parallel forces that have the same

magnitude and are separated by a perpendicular distance.

The resultant force of a couple is zero.

A couple causes rotation.


COUPLE MOMENT (page 2)

• Couple moment

The couple moment

!M

O is the sum of the moments of the two forces

that form the couple:

!M

O=!rB!!F +

!rA! ("!F)

= (!rB!!rA) "!F

That is,

!M

O=!r !!F where

!r =!rB!!rA

• Free vector

The couple moment depends on the position vector !r directed from A to

B and not on the position of the arbitrary point O.

The subscript O on

!M

O will therefore be omitted:

!M =

!r !!F

!M is a free vector. It can act at any point on a body.

• Direction

!M is perpendicular to the plane containing

!F and !

!F with direction

given by the right-hand rule (see Figure 4-27).

• Magnitude

M = Fd d = rsin!

! is the angle between the tail of !r and the tail of

!F .

d is the lever arm for the couple.



• Equivalent couples

!M1 and

!M2 produce the same turning if they are parallel and

F1d1= F

2d2

• Free-body diagrams

!M is drawn on free-body diagrams rather than the individual couple

forces !F and !

!F .


COUPLE

• Couple

A couple is defined as two anti-parallel forces that have the same

magnitude and are separated by a perpendicular distance.

The resultant force of a couple is zero.

A couple causes rotation.



• Couple moment

The couple moment

!M

O is the sum of the moments of the two forces

that form the couple:

!M

O=!rB!!F +

!rA! ("!F)

= (!rB!!rA) "!F

That is,

!M

O=!r !!F where

!r =!rB!!rA

• Free vector

The couple moment depends on the position vector !r directed from A to

B and not on the position of the arbitrary point O.

The subscript O on

!M

O will therefore be omitted:

!M =

!r !!F

!M is a free vector. It can act at any point on a body.

• Direction

!M is perpendicular to the plane containing

!F and !

!F with direction

given by the right-hand rule (see Figure 4-27).

• Magnitude

M = Fd d = rsin!

! is the angle between the tail of !r and the tail of

!F .

d is the lever arm for the couple.



• Equivalent couples

!M1 and

!M2 produce the same turning if they are parallel and

F1d1= F

2d2

• Free-body diagrams

!M is drawn on free-body diagrams rather than the individual couple

forces !F and !

!F .

Section 4.7 Simplification of a Force and Couple System (page 160)

RESULTANT FORCE AND COUPLE MOMENT (page 1)

Section 4.7 Simplification of a Force and Couple System (page 160)

RESULTANT FORCE AND COUPLE MOMENT (page 2)

• Figure (a)

Figure (a) shows a rigid body acted on by forces

!F1

and

!F2 and by a

couple moment !M .

• Figure (b)

Figure (b) shows

!F1,

!F2 and

!M acting at O, which is any point not on

the line of action of the forces.

In addition to !M , there are two other couple moments at O:

(!M

O)1=!r1!!F1

(!M

O)2=!r2!!F2

These couple moments arise from moving

!F1

and

!F2

to O.

(“Move a force, add a couple.”)

• Figure (c)

Figure (c) shows the equivalent resultant force

!FR

and equivalent

resultant couple moment (!M

R)O

acting at O .

Equations for the resultant force and the resultant couple moment

!FR= !!F

(!M

R)O= !

!M + ! (

!r "!F)

Section 4.8 Further Simplification of a Force and Couple System

REDUCTION TO A WRENCH

Section 4.8 Further Simplification of a Force and Couple System

REDUCTION TO A WRENCH (page 2)

• Figure (a)

Figure (a) shows an equivalent resultant force

!FR

and equivalent

resultant couple moment (!M

R)O

acting at O as described in Section 4.7.

(!M

R)O

is written in terms of components:

(!M

R)O=!M"+!M

!

!M"

is parallel to

!FR

!M

! is perpendicular

!FR

Line b is perpendicular to the plane of

!FR

and (!M

R)O

!M

! points along line a

• Figure (b)

Figure (b) shows

!FR

at P.

!M

!has been cancelled by moving

!FR

to P

where

!rPO

!!FR= "!M

# (“move a force, add a couple”)

• Figure (c)

Figure (c) shows a wrench acting at P.

The resultant force

!FR

and resultant couple moment

!M" are parallel.

section 4.1 moment of a force (page 121)mattison/courses/phys170/p170-n4.pdf · section 4.1 moment...

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