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Let’s analyze force more deeply. Suppose a physics student sits at a table, working diligently on his homework ( [link] ). What external forces act on him? Can we determine the origin of these forces?

Figure a shows a person sitting on a chair with his forearms resting on a table. Force C in the upward direction and W in the downward direction, both having equal magnitude, act along the line of his torso. Force T is in the upward direction near the person’s forearms. Force F is in the upward direction near the person’s feet. Figure b shows the free body-diagram of C and W.
(a) The forces acting on the student are due to the chair, the table, the floor, and Earth’s gravitational attraction. (b) In solving a problem involving the student, we may want to consider the forces acting along the line running through his torso. A free-body diagram for this situation is shown.

In most situations, forces are grouped into two categories: contact forces and field forces . As you might guess, contact forces are due to direct physical contact between objects. For example, the student in [link] experiences the contact forces C , F , and T , which are exerted by the chair on his posterior, the floor on his feet, and the table on his forearms, respectively. Field forces, however, act without the necessity of physical contact between objects. They depend on the presence of a “field” in the region of space surrounding the body under consideration. Since the student is in Earth’s gravitational field, he feels a gravitational force w ; in other words, he has weight.

You can think of a field as a property of space that is detectable by the forces it exerts. Scientists think there are only four fundamental force fields in nature. These are the gravitational, electromagnetic, strong nuclear, and weak fields (we consider these four forces in nature later in this text). As noted for w in [link] , the gravitational field is responsible for the weight of a body. The forces of the electromagnetic field include those of static electricity and magnetism; they are also responsible for the attraction among atoms in bulk matter. Both the strong nuclear and the weak force fields are effective only over distances roughly equal to a length of scale no larger than an atomic nucleus ( 10 −15 m ). Their range is so small that neither field has influence in the macroscopic world of Newtonian mechanics.

Contact forces are fundamentally electromagnetic. While the elbow of the student in [link] is in contact with the tabletop, the atomic charges in his skin interact electromagnetically with the charges in the surface of the table. The net (total) result is the force T . Similarly, when adhesive tape sticks to a piece of paper, the atoms of the tape are intermingled with those of the paper to cause a net electromagnetic force between the two objects. However, in the context of Newtonian mechanics, the electromagnetic origin of contact forces is not an important concern.

Vector notation for force

As previously discussed, force is a vector; it has both magnitude and direction. The SI unit of force is called the newton    (abbreviated N), and 1 N is the force needed to accelerate an object with a mass of 1 kg at a rate of 1 m/s 2 : 1 N = 1 kg · m/s 2 . An easy way to remember the size of a newton is to imagine holding a small apple; it has a weight of about 1 N.

We can thus describe a two-dimensional force in the form F = a i ^ + b j ^ (the unit vectors i ^ and j ^ indicate the direction of these forces along the x -axis and the y -axis, respectively) and a three-dimensional force in the form F = a i ^ + b j ^ + c k ^ . In [link] , let’s suppose that ice skater 1, on the left side of the figure, pushes horizontally with a force of 30.0 N to the right; we represent this as F 1 = 30.0 i ^ N . Similarly, if ice skater 2 pushes with a force of 40.0 N in the positive vertical direction shown, we would write F 2 = 40.0 j ^ N . The resultant of the two forces causes a mass to accelerate—in this case, the third ice skater. This resultant is called the net external force     F net and is found by taking the vector sum of all external forces acting on an object or system (thus, we can also represent net external force as F ):

F net = F = F 1 + F 2 +

This equation can be extended to any number of forces.

In this example, we have F net = F = F 1 + F 2 = 30.0 i ^ + 40.0 j ^ N . The hypotenuse of the triangle shown in [link] is the resultant force, or net force. It is a vector. To find its magnitude (the size of the vector, without regard to direction), we use the rule given in Vectors , taking the square root of the sum of the squares of the components:

F net = ( 30.0 N ) 2 + ( 40.0 N ) 2 = 50.0 N .

The direction is given by

θ = tan −1 ( F 2 F 1 ) = tan −1 ( 40.0 30.0 ) = 53.1 ° ,

measured from the positive x -axis, as shown in the free-body diagram in [link] (b).

Let’s suppose the ice skaters now push the third ice skater with F 1 = 3.0 i ^ + 8.0 j ^ N and F 2 = 5.0 i ^ + 4.0 j ^ N . What is the resultant of these two forces? We must recognize that force is a vector; therefore, we must add using the rules for vector addition:

F net = F 1 + F 2 = ( 3.0 i ^ + 8.0 j ^ ) + ( 5.0 i ^ + 4.0 j ^ ) = 8.0 i ^ + 12 j ^ N

Check Your Understanding Find the magnitude and direction of the net force in the ice skater example just given.

14 N, 56 ° measured from the positive x -axis

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View this interactive simulation to learn how to add vectors. Drag vectors onto a graph, change their length and angle, and sum them together. The magnitude, angle, and components of each vector can be displayed in several formats.


  • Dynamics is the study of how forces affect the motion of objects, whereas kinematics simply describes the way objects move.
  • Force is a push or pull that can be defined in terms of various standards, and it is a vector that has both magnitude and direction.
  • External forces are any outside forces that act on a body. A free-body diagram is a drawing of all external forces acting on a body.
  • The SI unit of force is the newton (N).

Conceptual questions

What properties do forces have that allow us to classify them as vectors?

Forces are directional and have magnitude.

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Two ropes are attached to a tree, and forces of F 1 = 2.0 i ^ + 4.0 j ^ N and F 2 = 3.0 i ^ + 6.0 j ^ N are applied. The forces are coplanar (in the same plane). (a) What is the resultant (net force) of these two force vectors? (b) Find the magnitude and direction of this net force.

a. F net = 5.0 i ^ + 10.0 j ^ N ; b. the magnitude is F net = 11 N , and the direction is θ = 63 °

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A telephone pole has three cables pulling as shown from above, with F 1 = ( 300.0 i ^ + 500.0 j ^ ) , F 2 = −200.0 i ^ , and F 3 = −800.0 j ^ . (a) Find the net force on the telephone pole in component form. (b) Find the magnitude and direction of this net force.

Figure shows the coordinate axes, vector F1 at an angle of about 28 degrees with the positive y axis, vector F2 along the negative x axis and vector F3 along the negative y axis.
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Two teenagers are pulling on ropes attached to a tree. The angle between the ropes is 30.0 ° . David pulls with a force of 400.0 N and Stephanie pulls with a force of 300.0 N. (a) Find the component form of the net force. (b) Find the magnitude of the resultant (net) force on the tree and the angle it makes with David’s rope.

a. F net = 660.0 i ^ + 150.0 j ^ N ; b. F net = 676.6 N at θ = 12.8 ° from David’s rope

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Questions & Answers

Is there any calculation for line integral in scalar feild?
Sadia Reply
what is thrust
Aarti Reply
when an object is immersed in liquid, it experiences an upward force which is called as upthrust.
@Phanindra Thapa No, that is buoyancy that you're talking about...
thrust is simply a push
it is a force that is exerted by liquid.
what is the difference between upthrust and buoyancy?
The force exerted by a liquid is called buoyancy. not thrust. there are many different types of thrust and I think you should Google it instead of asking here.
hey Kumar, don't discourage somebody like that. I think this conversation is all about discussion...remember that the more we discuss the more we know...
thrust is an upward force acting on an object immersed in a liquid.
uptrust and buoyancy are the same
the question isn't asking about up thrust. he simply asked what is thrust
a Thrust is simply a push
how did astromers neasure the mass of earth and sun
papillas Reply
wats the simplest and shortest formula to calc. for order of magnitude
Distinguish between steamline and turbulent flow with at least one example of each
Aarti Reply
what is newtons first law
Theodore Reply
It state that an object in rest will continue to remain in rest or an object in motion will continue to remain in motion except resultant(unbalanced force) force act on it
Thanks Gerald Fokumla
Your welcome
it states that a body remains in its state of rest or uniform motion unless acted upon by resultant external force.
it that a body continues to be in a state of rest or in straight line in a motion unless there is an external force acting on it
derive the relation above
jeba Reply
formula for find angular velocity
Mark Reply
Why satellites don't fall on earth? Reason?
Abdur Reply
because space doesn't have gravity
satellites technically fall to earth but they travel parallel to earth so fast that they orbit instead if falling(plus the gravity is also weaker in the orbit). its a circular motion where the centripetal force is the weight due to gravity
Exactly everyone what is gravity?
Tebogo Reply
the force that attrats a body towards the center of earth,or towards any other physical body having mass
That force which attracts or pulls two objects to each other. A body having mass has gravitational pull. If the object is bigger in mass then it's gravitational pull would be stronger.For Example earth have gravitational pull on other objects that is why we are pulled by earth.
Gravity is the force that act on a on body to the center of the earth.
what are the application of 2nd law
Sana Reply
It's applicable when determining the amount of force needed to make a body to move or to stop a moving body
coplanar force system
Alex Reply
how did you get 7.50times
Cole Reply
what is a frame of reference
The system of geometric axis in relation to which measurement of Size, Position, or , Motion can be made. It has two types; 1) Inertial Reference Frame 2) Non Inertial Reference Frame
what is science
Lloyd Reply
What is Matter?
what is black body radiations?
matter is anything having some mass and occupies some volume
a black body radiation it the radiation that absorbs all the EM radiation
is anything that occuple space
I'm new here...I wana askng u how can I prepare any typ of test of atomic energy
What is inertia of bank curve
I wanna ask the different between coloumbs law and gravitational law of force
how do you got 27.8 m/s? please explain
Yusuf Reply
Practice Key Terms 6

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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