by Julia Logan

editing claimed by Leila Bond

Definition

A Mathematical Model

In order to calculate net force, all EXTERNAL forces acting on a system are added together. The mathematical definition is

[math]\displaystyle{ Fnet = F1 + F2 + F3... }[/math]

Additionally,

[math]\displaystyle{ Fnet = ma }[/math]
where m=mass of the object, and a = acceleration of the object.

This is a result of Newton's Second Law of motion. If there is a nonzero net force acting on an object, that object is accelerating (not traveling at a constant velocity). Interestingly, there is zero net force acting on an object if its velocity is constant. This seems counter-intuitive (surely something is causing the object to move!) but makes sense in the context of Newton's Second Law. The forces are balanced (sum to zero) if there is no acceleration, despite any movement that may be happening.

A Computational Model

Net force is an essential component of the Momentum Principle! We can use the Momentum Principle in vpython to update the position of a moving object. But first, we have to find net force.

Tracing the path of a ball/spring model in vpython using net force and the momentum principle.

  #1 Fspring = -k*s
  #2 Fgravmag = mball * g
  #3 Fgrav = Fgravmag * vector(0,-1,0)
  #4 Fnet = Fspring+Fgrav
  #5 pball = pball + Fnet * deltat
  #6 vball = pball / mball
  #7 ball.pos=ball.pos+vball*deltat

Here, the spring force and the gravitational force are found using formulas (lines 1-3). Then, they are added together to get the net force on the object (in this case a ball, line 4). The net force is then used in the update form of the momentum principle (line 5). In line 6 the velocity is updated, and line 7 the position is updated. Without net force calculations, tracing an object's path would be impossible.

Simple Example

When calculating net force, it is most useful to construct a free body diagram. A free body diagram is a physical representation of the external forces applied to a system. Often, arrows are used to represent forces. In this example, forces are displayed acting on a box.

The first two boxes' net force can be easily calculated by adding the two forces acting on them, respectively. With down being in the negative y direction, for the first box we have:

[math]\displaystyle{ Fnet = (0, 1200, 0) N + (0, -800, 0) N }[/math]
[math]\displaystyle{ Fnet = (0, 400, 0) N }[/math]

And the second box:

[math]\displaystyle{ Fnet = (0, 600, 0) N + (0, -800, 0) N }[/math]
[math]\displaystyle{ Fnet = (0, -200, 0) N }[/math]

The third box has both x-component forces and y-component forces. It is important to separate these in solving for the net force.
y-components:
[math]\displaystyle{ Fnet,y = (0, 50, 0) N + (0, -50, 0) N }[/math]
[math]\displaystyle{ Fnet,y = (0, 0, 0) N }[/math]
x-components:
[math]\displaystyle{ Fnet,x = (-20, 0, 0) N + (0, 0, 0) N }[/math]
Put them together:
[math]\displaystyle{ Fnet = (-20, 0, 0) N }[/math]

More Complicated Example

There is a box, small enough to represented as a point, sitting on a slope inclined twenty degrees with respect to the horizontal. If its mass is 10 kg, what does the magnitude of the force of static friction (pointing up the hill) have to be for net force to be zero? In that case, what is the magnitude of the normal force? (All forces are measured in newtons, and the acceleration due to gravity [math]\displaystyle{ = g = 9.8 }[/math] [math]\displaystyle{ m/(s^2) }[/math]).

Let [math]\displaystyle{ f }[/math] = force of friction, [math]\displaystyle{ W }[/math] = weight of the object, and [math]\displaystyle{ N = }[/math] normal force. (The pictures are not to scale.)

First, we have to identify our coordinate axes. We could have the typical horizontal x-axis and vertical y-axis (and we know where to put [math]\displaystyle{ θ }[/math] using similar triangles):

But, that leaves us with two forces that have to be decomposed into x- and y- components. It makes more sense to draw the axes rotated twenty degrees counterclockwise, like this:

[math]\displaystyle{ W }[/math] is easy to find, since it's just mass times acceleration due to gravity, both of which are known quantities.

[math]\displaystyle{ W = mg = (10 }[/math] [math]\displaystyle{ kg)*(9.8 }[/math] [math]\displaystyle{ m/(s^2)) = 98 }[/math] [math]\displaystyle{ newtons }[/math]

Now we are ready to solve for the frictional force. It acts in the positive x direction, and we know that the net force in the x direction has to be zero, so we have to identify the other forces that act in the positive or negative x direction. In this case, it's only a component of the weight, which points in the negative x direction. Using simple trigonometry, we find that that is

[math]\displaystyle{ -W*sin(θ) }[/math]

So we can write our equilibrium equation like this:

[math]\displaystyle{ Fnetx = f - W*sin(θ) = 0 }[/math]

Then, all we have to do is solve for [math]\displaystyle{ f }[/math], since [math]\displaystyle{ W }[/math] and [math]\displaystyle{ θ }[/math] are known quantities.

[math]\displaystyle{ f = W*sin(θ) = (98 }[/math] N[math]\displaystyle{ )*sin(20°) = 33.51797405 }[/math] N, which is about equal to [math]\displaystyle{ 33.5 }[/math] N.

Now we have to solve for the normal force. It acts in the positive y direction, and we know that the net force in the y direction has to be zero, so we have to identify the other forces that act in the positive or negative y direction. In this case, the only one is the other component of the weight, which points in the negative y direction. Using simple trigonometry, we find that that is

[math]\displaystyle{ -W*cos(θ) }[/math]

So we can write our equilibrium equation like this:

[math]\displaystyle{ Fnety = N - W*cos(θ) = 0 }[/math]

Then, all we have to do is solve for [math]\displaystyle{ N }[/math], since [math]\displaystyle{ W }[/math] and [math]\displaystyle{ θ }[/math] are known quantities.

[math]\displaystyle{ N = W*cos(θ) = (98 }[/math] N[math]\displaystyle{ )*cos(20°) = 92.08987684 }[/math] N, which is about equal to [math]\displaystyle{ 92.1 }[/math] N.

Connectedness

Net force is one of the building blocks of Intro Physics, and I would assume all of physics. It's really important for all motion-related topics, specifically Curving Motion. A net force due to the gravitational pull of the Sun in the perpendicular direction is how the Earth revolves around the Sun -- why we have days and nights and years! We can see from the derivative form of the Momentum Principle that any change in momentum is due to a nonzero net force acting on a system. All changes in motion can be attributed to a net force acting in some direction.

[math]\displaystyle{ dP/dt=Fnet }[/math]

History

Some of the earliest records of humans' musings about force come from Aristotle. Aristotle observed the natural world and made assumptions and equations based on what he saw. He described all motion as being either "natural"--circular and infinite OR finite, up and down, in a straight line--or "violent." Aristotle had issues with projectile motion as he could not reconcile the continuing movement of the object with the lack of force being applied. He ended up concluding that the air provides a simultaneous resistant and accelerating force to the object.

These ideas were challenged by Renaissance men such as John Philoponus, John Buridan, and Oresme, with a clear understanding of the conservation of linear momentum not arriving to the world until Descartes. The most famous face we associate with forces in Physics is Sir Isaac Newton, and he certainly played a huge role in the understanding of net force and momentum. However, he initially believed in the idea of impetus, that a projectile has a certain internal force that keeps it moving, and also the idea of transfer, that objects give up parts of their force during a collision. Though he was wrong, these ideas are not at all silly--in fact they seem quite logical--and he later corrected them after a series of experiments. Thus, the second law of motion was established and with it the relationship between force and motion.

References

Cardenas, Richard. "What is Net Force? - Definition, Magnitude & Equations." Web. 30 Nov 2015. http://study.com/academy/lesson/what-is-net-force-definition-magnitude-equations.html

"Net Force." Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. Web. 30 Nov 2015. https://en.wikipedia.org/wiki/Net_force

The Physics Classroom. "Determining the Net Force." Web. 30 Nov 2015. http://www.physicsclassroom.com/class/newtlaws/Lesson-2/Determining-the-Net-Force

Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. Modern Mechanics. Hoboken, NJ: Wiley, 2011. Print.

Stinner, Arthur. "The story of force: from Aristotle to Einstein." Physics Education. 1994. Web. 30 Nov 2015. http://www.arthurstinner.com/stinner/pdfs/1994-storyofforce.pdf

Net Force

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