# Physics: Newton's Laws

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This calculator provides fundamental Newtonian physics equations related to the Forces and Newton's Laws.

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# Forces

Forces come in a number of varieties:

• contact force - the force of one object making direct contact and pushing on another body. A surface typically exerts a "normal" force (meaning perpendicular) on an object sitting on a surface and this is a contact force.
• tension force - the force exerted by a body pulling on another body
• gravitational force - the force exerted "long range" on a body by another body. Gravitational force is a characteristic attribute of all mass.
• magnetic force - the force exerted "long range" on a body due to magnetic fields
• friction force - the force acting parallel to a surface which opposes one object sliding on another
• and many other kinds of forces

When multiple forces act on an object at the same point, they have the same affect as the vector sum of the multiple forces:

[1] vecF_("net") = vecF_1 +vecF_2 + ... + vecF_n  1

This means we can calculate a net force in any particular scenario that described the force on any one object in that scenario.

In two dimensions two forces can be superimposed using the component forces.  A component of the net force can for example be expressed as:

[2] vecF_("net-x") = vecF_"(1x)" +vecF_"(2x)" + ... + vecF_"(nx)"

[3] vecF_("net") = (vecF_("ax") + vecF_("bx")) hati + (vecF_("ay") + vecF_("by")) hatj  2

and the magnitude of the next force can be determined from the component forces:

[4]F_("net") = sqrt(F_("net-x")^2 + F_("net-y")^2 + F_("net-z")^2)

# Newton's Laws

Newton's Laws of Motion state three basic physics principles and provide the foundation of classical mechanics:

• 1st Law: If the net force on a body is zero, the body will move with constant velocity and no acceleration.
• 2nd Law: Net external force on a body causes the body to accelerate in the direction of the net force.
• 3rd Law: The forces of body A on another body B is equal to the force that body B exerts on body A but is in the opposite direction. For every action there is an equal and opposite reaction.

## Newton's First Law

Newton's First Law of Motion states "When viewed in an inertial reference frame, an object either remains at rest or continues to move at a constant velocity, unless acted upon by an external force."3  The tendency of matter to remain at a constant velocity is called inertia.  A reference frame in which this law is valid is termed an inertial reference frame. 4

Mathematically, Newton's First Law is expressed for a body in equilibrium as:

[5] sum vecF = 0 5, which [4] then implies: sum F_x = 0 and sum F_y = 0 6

We see this law in our everyday experience when objects sliding on ice seem to want to keep going.  The frictional force working to slow them down is less and a car on ice will slide and slide.  In outer space, well beyond the slowing effects of the atmosphere, a rocket once put in motion will continue in motion indefinitely.  It's amazing that in Newton's day he was able to perceive a Law that applies most obviously in space travel.7

## Newton's Second Law

A net force will cause a body to accelerate in the direction of the net force.  And the magnitude of the acceleration is directly proportional to the net force on the body, which then implies that the ratio of the net force to the magnitude of the acceleration is constant and equal to the mass.  We see this in our everyday experience when we push a rolling object.  Our net force on that object causes the object to accelerate.  The acceleration we cause is less with a body of greater mass.

[5] m = |sum vecF| /a  8  (see the Force/Mass/Acceleration equation on the Forces tab.)

In SI units this equation is expressed as:

[6] 1 kg = (1 N) / (1 "meter"/s^2)  (see the Force Conversions equation on the Newton's Second Law of Motion tab)

If same net force is applied to two masses, m_1 and m_2, then [5] implies that:

[7] m_2/m_1 = a_1 / a_2  (see the equation Mass From Equal Force on the Newton's Second Law of Motion tab)

If we re-state Newton;s Second Law in vector form, force and acceleration are related as:

[8] sumvecF = m* veca  [ see the equation:  Force ]

## Newton's Third Law

newton's third law describes the fact that when one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.  As a Law of the universe this applies in ALL cases.

When we set-up physics descriptions to explain combinations of forces, we can always add the vector forces up so that the sum of forces is zero.  This is the mathematical manifestation of equal and opposite forces.

An example of this would be you standing a flat surface.  When you stand still, you can feel your weight on the soles of your feet.  You feel yourself resting there.  What your nerves on the bottom of your feet are sensing is the compression caused by the force of the Earth pushing back against your weight pushing down.  The gravitational force of the Earth's mass pulls your perpendicular to the local surface  directly toward the center of the Earth's mass.  At the same time, the Earth pushes back against your weight with an equal amount of force.

Because these two forces, your weight and the Earth pushing back are equal as you stand still, you don't move.  You feel the force of the Earth pushing against your weight but you and the Earth remain, for that instant, in equilibrium.

# References

1. ^ Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 110, eq 4.1
2. ^ Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 110, eq 4.2
3. ^ http://en.wikipedia.org/wiki/Newton%27s_laws_of_motion
4. ^  http://en.wikipedia.org/wiki/Inertial_frame_of_reference
5. ^ Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 112, eq 4.3
6. ^ Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 112, eq 4.4
7. ^ Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 112, eq 4.4
8. ^ Young, Hugh and Freeman, Roger.  University Physics With Modern Physics.  Addison-Wesley, 2008. 12th Edition, (ISBN-13: 978-0321500625 ISBN-10: 0321500628 ) Pg 116, eq 4.5

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