How do Forces Add?
So far we have only considered the influence of one unbalanced force on an object and we found that the effect was to accelerate it. If the same force is applied to the same mass we always measure the same acceleration. Of course, everything on earth has many forces acting on it. However, if the object is not accelerating (a at rest or moving but not changing its speed or direction) then it is assumed that the net effect of the various forces is zero.
The question of how forces combine is an experimental one. We can make reasonable guesses but those guesses (hypotheses) must be tested by experiment. There is no way one can figure it out by just thinking. For example, let’s say that we have two identical springs. Each spring can be tested individually to verify that it gives the same acceleration to an object of mass m when they are extended by a certain amount. Now hook both springs to the object and pull in the same direction until both springs are extended the same amount as when they were used individually. It seems reasonable to call this twice the force and, therefore, expect the acceleration to be double what it was with one spring. But is it?
Another reasonable guess would be that when two forces are pulling in opposite directions, the net effect is zero force and zero acceleration. Is it?
Finally, if two forces are pulling in different directions, do we find the resultant by adding them like vectors?
These questions are illustrated in the figure. Experimental tests have been done. You may have a chance to do some in the lab. Very precise tests can be done with well designed frictionless pucks and a special spring shaped like a hoop. When the hoop is distorted from a circle it is pulling on the object to which it is attached with what we will call the “Standard Force.” If two hoops have the same shape then, individually, they have the same accelerating effect on the frictionless puck. A photograph of the puck at 1 s intervals allows us to find its acceleration.
In the photo, the fricitonless puck is shown at 1 s intervals while it is being pulled by one Standard Force. The table below shows the position of the puck, its velocities and acceleration:
time
position
velocity
acceleration
(s)
(dm)
(dm/s)
(dm/s2)
0.00 0.85 0.61 1.00 1.46 0.30 0.91 2.00 2.37 0.27 1.18 3.00 3.55 0.27 1.45 4.00 5.00 0.23 1.68 5.00 6.68 0.24 1.92 6.00 8.60 0.21 2.13 7.00 10.73 average:
0.25
Two equivalent springs are attached to the puck in parallel. Is the acceleration twice what it was with just one spring?
The acceleration for two parallel springs can be gleaned from the position data:
time
position
velocity
acceleration
(s)
(dm)
(dm/s)
(dm/s2)
0.00 0.20 0.75 1.00 0.95 0.50 1.25 2.00 2.20 0.45 1.70 3.00 3.90 0.53 2.23 4.00 6.13 0.54 2.77 5.00 8.90 Average: 0.51
The third photo shows two of the springs pulling the puck at 60° to each other:
time
position
velocity
acceleration
(s)
(dm)
(dm/s)
(dm/s2)
0.00 0.10 1.10 1.00 1.20 0.45 1.55 2.00 2.75 0.40 1.95 3.00 4.70 Average: 0.425
If the forces add like vectors then vector sum is 2 cos(30°) = 1.73 times the force for one spring alone. Notice that the acceleration is increased by the same factor 1.73 × 0.25 m/s2 = 0.43 m/s2 . Therefore the vector method of addition seems to predict the correct acceleration. This agreement inspires us to have confidence in the simple assumption that forces add like vectors.
Postscript:
By the way, we started by assumming that no acceleration occurred when no force was applied. We have a picture to prove that too. You can analyze it yourself. Remember Constant velocity is consistent with Zero Force.
