8/12/2019 section 3.2 1-20-03

1/13

3.2 Moments in a Plane

As discussed in the last section, a rigid body has at most six degrees-of-

freedom. Three of them are translational and three are rotational. When the analysis iconfined to a plane, the rigid body has at most three degrees-of-freedom. Two are

translational and one is rotational. These rigid bodies are called planar rigid bodies.

planar rigid bodies, the forces act in the plane and momentsare produced by these

forces. The forces and moments maintain the body in a state of translational

equilibrium and rotational equilibrium. This section shows several different ways how

to mathematically express a moment acting on a planar rigid body. Then its shown

how to find the resultantmoment acting on a planar rigid body.

.

The Moment of a Force

!igure " # $ shows point A and force F. The force acts at point B. T

perpendicular distance between pointA and the line of action of the force is denoted

a. The momentabout pointAproduced by the force is defined as the product of t

forceF and the perpendicular distance a, written

%" # "& MA= aF.

The perpendicular distance a is called the moment arm. Therefore, in words, 'q. %

"& states that the moment about point A produced by the force F is equal to t

moment arm a multiplied by the force F.

As mentioned above, the analysis of a planar rigid body is confined to a plan

The rigid body can translate in two independent directions and rotate about an ax

perpendicular to the plane. To maintain a planar rigid body in equilibrium, you need

maintain its translational equilibrium as well as its rotational equilibrium. !orces te

to cause a rigid body to translate and moments tend to cause a body to rotate. (o

statics problems, forces maintain a body in translational equilibrium and mome

maintain a body in rotational equilibrium.

)otation in a plane is commonly referred to as about an axis. (imilarly, t

moment about pointA is also said to be the moment about an axis perpendicular thex

plane through pointA. In planar problems, the moment can be said to be about a po

or about an axis.

The unit of moment in the (I system is Newtonmeter, abbreviated *+m a

the unit of moment in the ( (ystem is poundft, abbreviated lbft. Anotcommonly used word for moment is torque.

)efer again to !ig. " # $. *otice that momentMA is positive when a andF

positive and when MA is positive that the moment tends to cause the body to rot

aboutA counter-clocwise. Thus, the sense of the moment in !ig. " # $ is said to

positive counter-clocwise. The sense of the moment could have also been taen to

positive clocwise, as shown in !ig. " # . The sense of the moment can be selected

be positive counter-clocwise or positive clocwise depending on the convention th

you set up.

Moment about PointA

Moment !rm

!igure " # /0 A visually pleasing and easilyrecogni1able situation in which a moment is

being applied to an ob2ect.

34/

!igure " # $0 A moment at point A is produced

by a force F. The moment is defined as theforceFmultiplied by the moment arm a.

!igure " # 0 The sense of the moment can be

positive counter-clocwise or positiveclocwise depending on the convention that

you set up.

Torque

Moment about an !xis

8/12/2019 section 3.2 1-20-03

2/13

'quation %" # "& can be calculated several different ways. In this sub-secti

three different, but similar, ways are shown. The next sub-section shows how

calculate a moment using rectangular components of position and force. )eferring

!ig. " # 5, the angle between the line of action of the force and the line between poi

A andB is denoted by . The distance between points A andB is denoted by r. T

moment arm is then related to the distance rby

%" # 6& a = r sin.

!rom 'qs. %" # "& and %" # 6&, the moment aboutAcan be written as

%" # /& MA= r F sin.

'quation %" # 6& states that the moment aboutA produced by the force F is equal

the distance r between pointsA andB multiplied by the force F multiplied by t

sine of the angle between them.

*ext, notice that the component of the force perpendicular to the line betwepointsA andBisf = F sin. Therefore, from 'q. %" # /&,

%" # $& MA= r f.

'quation %" # $& is a third way of expressing a moment. 'quation %" # $& states that tmoment aboutA produced by the force F is equal to the distance r between poin

A andB multiplied by the component of the force perpendicular to r.

As an example, loo at !ig. " # 7. *otice that a force F = 899 lb is bei

applied to the bar, " ft from pointA. The angle between lineAB and the line of force

is "9

:

. The moment about pointA is calculated three ways in !ig. " # 7. The differways of calculating the moment are very similar.

Finding a Moment "sing #ectangular $omponents

Another way to calculate a moment is to first express the position vector fro

point A to pointB and the force vector in terms of their rectangular components.

shown in !ig. " # 89, the position vector and the force vector, in terms of th

rectangular components, are

%" # & ,,; %iF%ir yxAB FFyx +=+=

where

%" # 5&

sin,cos

,sin,cos

FFFF

ryrx

yx ==

==

!rom !ig. " # 89, 'q. %" # /&, and using the trigonometric ident

sincoscossin&sin% = , the moment aboutA is written out as