6161103 4.1 moment of a force – scalar formation

Moment of a force about a point or axis – a

measure of the tendency of the force to

cause a body to rotate about the point or axis

4.1 Moment of a Force – Scalar Formation


cause a body to rotate about the point or axis

Case 1

Consider horizontal force Fx,

which acts perpendicular to

the handle of the wrench and

is located dy from the point O

� Fx tends to turn the pipe about the z axis

� The larger the force or the distance dy, the greater the turning effect

� Torque – tendency of



� Torque – tendency of

rotation caused by Fx

or simple moment (Mo) z

� Moment axis (z) is perpendicular to shaded

plane (x-y)

� Fx and dy lies on the shaded plane (x-y)

Moment axis (z) intersects



� Moment axis (z) intersects

the plane at point O

Case 2

Apply force Fz to the wrench

� Pipe does not rotate about z axis

Tendency to rotate about x axis



� Tendency to rotate about x axis

� The pipe may not actually

rotate Fz creates tendency

for rotation so moment

(Mo) x is produced

Case 2

� Moment axis (x) is perpendicular to shaded

plane (y-z)

F and d lies on the shaded plane (y-z)



� Fz and dy lies on the shaded plane (y-z)



Case 3

Apply force Fy to the wrench

� No moment is produced about point O

Lack of tendency to rotate � Lack of tendency to rotate

as line of action passes

through O



In General

� Consider the force F and the point O which lies in the shaded plane

� The moment M about point O, � The moment MO about point O,

or about an axis passing

through O and perpendicular

to the plane, is a vector quantity

� Moment MO has its specified

magnitude and direction



Magnitude� For magnitude of MO,

MO = FdO

where d = moment arm or perpendicular distance from the axis at point O to its line of action of the force

� Units for moment is N.m



Direction

� Direction of MO is specified by using “right hand rule”using “right hand rule”

- fingers of the right hand are curled to follow the sense of rotation when force rotates about point O



Direction

- Thumb points along the moment axis to give the moment axis to give the direction and sense of the moment vector

- Moment vector is upwards and perpendicular to the shaded plane



Direction

MO is shown by a vector arrow

with a curl to distinguish it from

force vector

Example (Fig b)

� MO is represented by the counterclockwise curl, which indicates the action of F



Direction

� Arrowhead shows the sense of rotation caused by F

Using the right hand rule, the � Using the right hand rule, the direction and sense of the moment vector points out of the page

� In 2D problems, moment of the force is found about a point O



Direction

� Moment acts about an axis perpendicular to the plane containing F and dcontaining F and d

� Moment axis intersects

the plane at point O



Resultant Moment of a System of

Coplanar Forces

� Resultant moment, MRo = addition of the � Resultant moment, MRo = addition of the moments of all the forces algebraically since all moment forces are collinear

MRo = ∑Fd

taking clockwise to be positive



Resultant Moment of a System of

Coplanar Forces

� A clockwise curl is written along the equation � A clockwise curl is written along the equation to indicate that a positive moment if directed along the

+ z axis and negative

along the – z axis



� Moment of a force does not always cause rotation

� Force F tends to rotate the beam clockwise about A with moment

MA = FdAMA = FdA

� Force F tends to rotate the beam counterclockwise about B with moment

MB = FdB

� Hence support at A prevents

the rotation



Example 4.1

For each case, determine the moment of the

force about point Oforce about point O



Solution

� Line of action is extended as a dashed line to establish moment arm d

� Tendency to rotate is indicated and the orbit is � Tendency to rotate is indicated and the orbit is shown as a colored curl

)(.5.37)75.0)(50()(

)(.200)2)(100()(

CWmNmNMb

CWmNmNMa

o

o

==

==



Solution

)(.0.21)14)(7()(

)(.4.42)45sin1)(60()(

)(.229)30cos24)(40()(

CCWmkNmmkNMe

CCWmNmNMd

CWmNmmNMc

o

o

=−=

==

=+=

o

o

)(.0.21)14)(7()( CCWmkNmmkNMe o =−=



Example 4.2

Determine the moments of

the 800N force acting on the

frame about points A, B, C frame about points A, B, C

and D.

4.1 Moment of a Force –Scalar Formation


Solution

Scalar Analysis

)(.1200)5.1)(800(

)(.2000)5.2)(800(

CWmNmNM

CWmNmNM A

==

==

Line of action of F passes through C

)(.400)5.0)(800(

.0)0)(800(

)(.1200)5.1)(800(

CCWmNmNM

mkNmNM

CWmNmNM

D

C

B

==

==

==



Example 4.3

Determine the resultant moment of the four

forces acting on the rod about point Oforces acting on the rod about point O



Solution

Assume positive moments acts in the +k

direction, CCW

FdM ∑=

)(.334

.334

)30cos34)(40(

)30sin3)(20()0)(60()2)(50(

CWmN

mN

mmN

mNmNmNM

FdM

Ro

Ro

=

−=

+−

++−=

∑=

o

o

6161103 4.1 moment of a force – scalar formation

Technology

n force

moment vector moment

moment forces

moment of theforce

d moment axis

moment axis x

force scalar formationcase

force scalar formationexample