Forces

Forces acting on objects are vectors that are characterized by not only a magnitude (e.g. pounds force
or Newtons) but also a direction. A force vector F (vectors are usually noted by a boldface letter) can be broken down into its components in the x, y and z directions in whatever coordinate system you’ve drawn ;

     

Where Fx, Fy and Fz are the magnitudes of the forces in the x, y and z directions and i, j and k are the unit vectors in the x, y and z directions (i.e. vectors whose directions are aligned with the x, y and z coordinates and whose magnitudes are exactly 1 (no units)).

Forces can also be expressed in terms of the magnitude and direction relative to the positive x-axis   in a 2-dimensional system Note that the function gives you an angle between whereas sometimes the resulting force is between +90˚ and +180˚ or between -90˚ and -180˚; in these cases you’ll have to examine the resulting force and add or subtract 180˚ from the force to get the right direction. 

    

Some  types  of  structures  can  only  exert  forces along  the  line  connecting  the  two  ends  of  the structure, but cannot exert any force perpendicular to that line.  These types of structures include ropes, ends with pins, and bearings.  Other structural elements can also exert a force perpendicular to the line (Figure 5).  This is called the moment of force (often shortened to just “moment”, but to avoid confusion with “moment” meaning a short period of time, we will use the full term “moment of force”) which is the same thing as torque. Usually the term torque is reserved for the forces on rotating, not  stationary,  shafts,  but  there  is  no  real  difference  between  a moment of  force and  a torque.

The distinguishing feature of the moment of force is that it depends not only on the vector force itself (Fi) but also the distance (di) from that line of force to a reference point A. (I like to call this distance the moment arm) from the anchor point at which it acts. If you want to loosen a stuck bolt, you want to apply whatever force your arm is capable of providing over the longest possible di. The line through  the  force Fi is  called  the line  of  action.   The  moment  arm  is  the  distance  (di again) between the line of action and a line parallel to the line of action that passes through the anchor point. Then the moment of force (Mi) is defined as

    

where Fi
is the magnitude of the vector F. Note that the units of Mo is force x length, e.g. ft lbf or N m.   This  is  the  same  as  the  unit  of  energy,  but  the two  have  nothing  in  common – it’s just coincidence.   So  one  could  report  a  moment  of  force  in  units  of  Joules,  but  this  is  unacceptable practice – use N m, not J.

Note that it is necessary to assign a sign to Mi. Typically we will define a clockwise moment of force
as positive and counterclockwise as negative, but one is free to choose the opposite definition – as long as you’re consistent within an analysis.

In order to have equilibrium of an object, the sum of all the forces AND the moments of force must
be zero. In other words, there are two ways that a 2-dimensional object can translate (in the x and y
directions) and one way that in can rotate (with the axis of rotation perpendicular to the x-y plane.)
So there are 3 equations that must be satisfied in order to have equilibrium, namely

  

Note that the moment of forces must be zero regardless  of  the  choice  of  the  origin (i.e. not just at the center of mass). So one can take the origin to be wherever it is convenient (e.g. make the moment
of one of the forces = 0.) Consider the very simple set of forces below:

      

Because of the symmetry, it is easy to see that this set of forces constitutes an equilibrium condition.
When taking moments of force about point ‘B’ we have:

     

But how do we know to take the moments of force about point B? We don’t. But notice that if we take the moments of force about point ‘A’ then the force balances remain the same and 

   

The same applies if we take moments of force about point ‘C’, or a point along the line ABC, or even a point NOT along the line ABC. For example, taking moments of force about point ‘D’,

    

The  location  about  which  to  take  the  moments  of  force can  be  chosen  to  make  the  problem  as simple as possible, e.g. to make some of the moments of forces = 0.

Example  of  “why  didn’t  the  book  just  say  that…?”  The  state  of  equilibrium  merely  requires  that  3 constraint equations are required. There is nothing in particular that requires there be 2 force and 1 moment of force constraint equations. So one could have 1 force and 2 moment of force constraint equations:

     

where the coordinate direction x can be chosen to be in any direction, and moments of force are taken about 2 separate points A and B. Or one could even have 3 moment of force equations:

    

Also, there is no reason to restrict the x and y coordinates to the horizontal and vertical directions. They can be (for example) parallel and perpendicular to an inclined surface if that appears in the problem. In fact, the x and y axes don’t even have to be perpendicular to each other, as long as they are  not  parallel  to  each  other,  in  which case ΣFx =  0  and ΣFy=  0  would  not  be  independent equations.

This is all fine and well for a two-dimensional (planar) situation, what about 1D or 3D? For 1D there is only one direction that the object can move linearly and no way in which it can rotate.  For 3D, there are three directions it can move linearly and three axes about which it can rotate. Table 1 summarizes these situations.