Chapter 4. Moment - the tendency of a force to rotate an object

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Chapter 4

Moment - the tendency of a

force to rotate an object

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Finding the moment -

2D Scalar Formulation

M

o

= F d

Moment

about 0

Magnitude

of force

Perpendicular

distance between

LOA and 0

Rotation is

clockwise or

counter clockwise

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Principle of Moments:

Sum the moments from the components instead

F

a b

d O

MO = F d and the direction is counter-clockwise, but finding d could be

tough … a b O F F _x

F _y MO = (FY a) – (FX b), because a and b are

easier to find (given)

The typical sign convention for a moment in 2-D is that counter-clockwise is considered positive.

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Moment Calculation:

Vector Analysis

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Moment Calculation:

Vector Analysis

While finding the moment of a force in 2-D is straightforward when you know the perpendicular distance d, finding the

perpendicular distances can be hard—especially when you are working with forces in three dimensions.

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Moment Calculation:

Vector Analysis

So a more general approach to finding the moment of a force exists. This more general approach is usually used when

dealing with three dimensional forces but can also be used in the two dimensional case as well.

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Moment Calculation:

Vector Analysis

So a more general approach to finding the moment of a force exists. This more general approach is usually used when

dealing with three dimensional forces but can also be used in the two dimensional case as well.

This more general method of finding the moment of a force involves a vector operation called the cross product of two vectors.

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Cross Product (§4.2)

In general, the cross product of two vectors A and B results in another vector, C , i.e.,

C = A X B. The magnitude and direction of

the resulting vector can be written as C = A X B = A B sin uC

As shown, uC is the unit vector

perpendicular to both A and B vectors (or to the plane containing the A and B vectors).

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Cross Product

The right-hand rule is a useful tool for determining the direction of the vector resulting from a cross product.

Calculating dot products between unit vectors is key in BME201:

i X j = k j X k = i k X i = j

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Determinants

Also, the cross product can be written as a determinant.

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Determinants

Also, the cross product can be written as a determinant.

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Calculate moment about o using MO = r X F where r is the

position vector from point O to any point on the line of action of F

Vector Moment

Calculation (§4.3)

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r is vector to any point

on LOA

Find the moment of the weight about ‘o’ ...

~

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2D Example

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2D Example

Find the moment of the force about O

y

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2D Example

y

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2D Example

y

x

Fx

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2D Example

y

x

Fy = – 100 (3/5) N Fx = 100 (4/5) N

Fx

Fy

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2D Example

y

x

Fy = – 100 (3/5) N Fx = 100 (4/5) N + MO = {– 100 (3/5)N (5 m) – (100)(4/5)N (2 m)} N·m = – 460 N·m

Fx

Fy

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Principle of

Transmissibility

A force may be placed anywhere on its line of

action without altering the mechanical analysis for

a rigid body

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Principle of

Transmissibility

A force may be placed anywhere on its line of

action without altering the mechanical analysis for

a rigid body

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Principle of

Transmissibility

A force may be placed anywhere on its line of

action without altering the mechanical analysis for

a rigid body

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Principle of

Transmissibility

A force may be placed anywhere on its line of

action without altering the mechanical analysis for

a rigid body

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Principle of

Transmissibility

A force may be placed anywhere on its line of

action without altering the mechanical analysis for

a rigid body

=

y

x

point o

y

x

point o

y

x

point o

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Apply the Principles to

Reduce Work!

F

L1

L2

B

A

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Moments in 3-D can be calculated using scalar (2-D) approach but it can be difficult and time consuming

Must consider every component’s moment about every axis and get signs

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3D Example

Find the moment

of F about point O

rBO = {0 i + 3 j + 1.5 k} m

Vector analysis:

MO = rBO X F = {25.5 i - 9 j + 18 k} N·m

To the board ...

{0 i + 3 j + 1.5 k} X {-6 i + 3 j + 10 k}

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3D Example

Find the moment

of F about point O

Scalar analysis

= {25.5 i - 9 j + 18 k} N·m {3(10) – 1.5(3)} i – 1.5( 6) j + 3(6) k MO = MO =

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3D Example

Find the moment

of F about point O

Scalar analysis

= {25.5 i - 9 j + 18 k} N·m {3(10) – 1.5(3)} i – 1.5( 6) j + 3(6) k MO =

RHR

MO =

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Attention Quiz

If

M = r X F

, then what will be the value of

M •

r

?

A) 0! ! B) 1

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Attention Quiz

If

M = r X F

, then what will be the value of

M •

r

?

A) 0! ! B) 1

C) r

2

_{F! D) None of the above}

10 N

3 m P 2 m 5 N

Using the CCW direction as positive, the net moment of the two forces about point P is

A) 10 N !m B) 20 N !m C) - 20 N !m D) 40 N !m E) - 40 N !m

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Example 2

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4.43

Find the moment

produced by

F

A