1. TODAY’S TOPIC: Conditions of Equilibrium (i.e. Statics)
How do we define equilibrium for a system ?
What conditions ensure equilibrium of a system (i.e.
design it wrong and you’ll hear a lot of ‘static’) ?
A force is a force of course of course unless of course
its just a moment.
‘Flatlanders’ or seeing a problem two-dimensionally.
The ubiquitous ‘free-body diagram’.

2. Joe's Diner This is the question first posed to
me by the statics prof at UCSB
when I announced to him that I
would be his TA for the quarter:
45°
30°
Joe's Diner
What is the tension of in each
chain supporting the restaurant
sign? The sign weighs 20 kg.
The sign may has well have read ‘HELP!!!’. My ‘on the spot’
recall of Mechanics had left me hanging. How would you have
handled this if you were in my shoes?? By the end of this lecture
we will be able to handle this problem, as well as many other
more difficult problems.

3. First recall our definition of a system
First recall our definition of a system. A mechanical system is defined as a body or group of bodies that can be isolated from all other bodies. This system may be a single body or a combination of connected bodies and may be rigid or non-rigid. The state of the defined system is then determined by the external forces that are acting upon it.
The subject of statics deals with systems that are in equilibrium. Note that equilibrium connotes balance (a balance of what?) and observationally implies that a system is not undergoing acceleration. A system is in equilibrium if it is at rest or in a state of uniform motion. A system at rest is said to be in static equilibrium.

4. From a short-term perspective many objects in nature appear to be in equilibrium such as this lava shelf on the Big Island of Hawaii formed by the flow of lava from the active volcano Kilauea.

5. However, nature is always at work and a combination of storms and earth tremors may suddenly change a system in static equilibrium into a dynamic system with dire consequences to man-made structures. This hillslide in La Conchita was the result of six inches of rainfall in two hours. Nine houses were damaged.

6. The engineer’s job in designing structures is to ensure that a system subject to external forces remains in static equilibrium over the anticipated lifespan of that system. The study of statics is the first step in this process. Statics is a stepping-stone to other studies such as strength of materials, vibrational analysis and fluid mechanics.
Equilibrium requires a balance of forces such that the acceleration of a system is zero.
The first condition of equilibrium is then
FNET = 0
(note that this is the net external force acting on the system)

7. Force is of course a vector quantity and we must sum the forces in the x, y and z directions when applying the first condition, or
Fx = 0, Fy = and Fz = 0
Is the first condition enough to ensure equilibrium? Consider a system that has only a couple acting on it. Recall that these forces are equal in magnitude and opposite in direction and thus the net external force on the system is zero. F d

8. In this case, the linear acceleration is zero, however there will be a change in angular motion due to the couple and the system will be in equilibrium only if a counter-couple of the same magnitude is applied. The second condition needed is
MNET = 0
(recall that in engineering the letter M is used as the symbol for torque and is called a moment). Again, this is the net external torque applied to the system.. Since there are three principle axes of rotation we can write the above vector equation as
Mx = 0,
My = 0 and
Mz = 0

9. In many designed systems external forces act only in 2-D and only three equations are needed for analysis:
Fx = 0, Fy = 0 and
Mo = 0
To the right are some common components of designed systems and their associated force components. Please study this figure carefully as they are essential in solving statics problems.

10. “The free-body diagram is the most important single step in the solution of problems in mechanics” - Jerry Meriam
Steps in constructing a free-body diagram:
Clearly decide which body or bodies is to be isolated and choose it such that the desired unknown(s) can be determined .
Draw a diagram of the isolated body that completely represents its external boundary.
Draw using vector arrows all of the known external forces that are applied to the isolated body and include the magnitude along the arrow. Unknown forces are also drawn with vector arrows and associated vector symbols. Try and guess the sense of the force, if you are wrong the answer will come out negative. Also, moments are included in the diagram with a a clockwise or counter-clockwise symbol followed by its magnitude.
Choose a convenient coordinate system and needed dimensions.

11. Here are some sample free-body diagrams.

12. Now let’s see about solving the problem presented previously.
Free-body diagram: T1 T2 y
30°
45°
Fx = 0 : T1sin30° - T2sin45° = 0 x
Fy = 0 : T1cos30° + T2cos45° - Mg = 0
Note that we have two equations and two unknowns and can
now solve for T1 and T2 by algebraic means. Mg
M=20 kg
T1 = 203 N
T2 = 144 N
Check by x-component: 203N* sin30° = 102 N, N* sin45° = 102 N
Another method is to note that the resultant of the vector sum is zero. T1 T2 Mg
Using the Law of Sines we have 45
Again T1 = 203 N and
T2 = 144 N
105
T T Mg = = 30
sin45° sin30° sin105°

13. Let’s try another problem
Let’s try another problem. Please take a couple of minutes to set it up on your own.
Construct a free-body diagram.
Solve for the unknown forces from the given data.
Check your answer.
Try and think of an alternative solution to the problem.