They are told that after identifying all forces and drawing a free-body diagram, they should start all problem solutions with the three statements:
....SUM(Fx) = 0...SUM(Fy) = 0...SUM(T) = 0.
Once they buy into this message, they have no difficulty solving static equilibrium problems. Let's look at a couple of examples.
NOTE: Because of the limitations of the text editor, I do use some rather unusual notation. That notation is summarized in the Ezine article "Teaching Rotational Dynamics".
Problem. The upper end of a uniform ladder of weight 300 N is placed against a wall. The other end of the ladder rests on the floor. The angle between the ladder and the floor is 45. The floor exerts a frictional force on the lower end of the ladder. However, the upper end of the ladder rests against a slippery (no friction) vertical wall. (a) Determine the forces of the wall and of the floor on the ladder. (b) What minimum coefficient of static friction Us is necessary to keep the lower end of the ladder from slipping?
Analysis. (a) The ladder is supported by the floor (normal force S and frictional force f). The ladder's upper end is resting against the smooth wall (normal force P with no frictional force). The weight of the ladder is 300 N and acts at its geometric center. We'll assume that the length of the ladder is L. We calculate torques around the axis through the base of the ladder (where it touches the floor). The moment arms of the force P, the 300-N weight, the force S, and the friction force f are Lsin45, Lcos45/2, 0, and 0, respectively. With the help of a free-body diagram, we apply the conditions for static equilibrium:
...................Conditions for Static Equilibrium
....f - P = 0............S - 300 N = 0........P(Lsin45) - (300 N)(Lcos45/2) = 0
The length L is a common factor in the torque equation and can therefore be cancelled. This leaves an equation for the single unknown force P. Solving for P, we find
.....................................P = 150 N.
Now the two force equations are easily solved for f and S:
.............f = P = 150 N........................S = 300N.
(b) The ladder does not slip as long as the maximum force of static friction (UsS) is greater than the actual force of static friction (f):
...............................UsS > f.
......................Us(300 N) > 150 N,
so.............................Us > 0.50.
Problem. The left end of a uniform horizontal beam of mass M = 200 kg is connected by a hinge to the wall, and the right end of the beam is tied to a light cable. The other end of the cable is attached to the wall so that it is oriented at 30 to the horizontal. The length of the beam is L= 8.0 m. A man of mass m = 80 kg walks along the beam from the hinge outward. If the breaking tension of the cable is 2500 N, what distance l along the beam can the man walk before the cable snaps?
Analysis. We'll investigate the beam. The beam is touching: (i) the hinge (forces Hx and Hy), (ii) the man's feet (weight mG downward), (iii) the cable (tension T). (iv) The weight of the beam is MG. With the help of a free-body diagram, we apply the conditions of static equilibrium to the beam. With torques taken around the beam's left end through the hinge, we have
............................................Conditions for Static Equilibrium
Since we want the point where the cable breaks, we set T = 2500 N. After inserting this along with m = 80 kg, M = 200 kg, L = 8.0 m, and G = 9.8 m/s into the three equations, we find easily that
..............................Hx = 2165 N...............Hy = 1494 N...............l = 2.8 m.
When the man reaches a point 2.8 m from the hinge, the cable breaks.
Every static equilibrium problem I cover is approached in the same way. I never solve a problem without starting with the three conditions of equilibrium. Like every topic I cover, I do my best to emphasize how easily physics problems are solved if they are just approached in terms of fundamental ideas.
Dr William Moebs is a retired physics professor, who taught at two Universities: Indiana-Purdue Fort Wayne and Loyola Marymount University. You can see hundreds of examples illustrating how he emphasizes fundamental principles by consulting PHYSICS HELP.
