Classical Mechanics/Newtonian Physics

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Newton's three laws of motion are the basis of classical mechanics, so they are presented here for review and completeness.

The First Law: Uniform Motion[edit]

An object in an inertial frame of reference having uniform, linear motion will remain that way unless acted upon by a net force.

or, Newton's First law may also be stated as : "In an inertial frame, if a body is at rest or in uniform motion, then it will continue to do so unless it is acted upon by an external force." Please note that the first law is a "fundamental law of nature". As such, it is not derivable from other laws, e.g. by putting F=0 in 2nd Law, and hence getting a=0, one can not say that the first law can be "mathematically" derived from 2nd Law. This is because the 2nd Law is applicable ONLY in inertial frames for which the first law gives a characterization. Hence, an inertial reference frame is one in which the motion is self sustaining, that is, an object in a state of rest or uniform linear motion will remain in that state forever unless acted upon by any external agency. Once, you have identified your inertial frame, it's only then that you can sensibly talk about the second law.

Note that an object at rest is a special case in an inertial frame of an object in uniform, linear motion, that being a object with zero velocity.

The Second Law: Force and Momentum[edit]

The rate of change of momentum of an object is equal to the net force applied on it.

Mathematically stated, the second law reads

 \textbf{F} = \frac{d\textbf{p}}{dt} = \frac{d(m\textbf{v})}{dt},

where

\textbf{F} is (net) force
\textbf{p} is momentum
m \, is mass
\textbf{v} is velocity
t \, is time

In many problems in mechanics mass is constant, in which case the law can be restated in its best known form,

 \textbf{F} = \frac{d(m\textbf{v})}{dt} = m \frac{d \textbf{v} }{dt} = m \textbf{a}

where \textbf{a} is acceleration. Note that the second law actually implies the first: where net force is zero (and mass is unchanging), acceleration must be zero, and so calculus tells us that the velocity must be constant.

The Third Law: Action and Reaction[edit]

For every action, there is an equal and opposite reaction.
All forces occur in pairs equal in magnitude and opposite in direction.

Suppose we have bodies A and B, Mathematically stated, the third law reads

 \textbf{F}_{AB} = -\textbf{F}_{BA} ,

where

\textbf{F}_{AB} is the force exerted by A upon B
\textbf{F}_{BA} is the force exerted by B upon A

Newton's third law is also known as the Weak Law of Action and Reaction.[1] This is to distinguish it from the Strong Law of Action and Reaction, which has the additional requirement that the forces also be central, meaning that they act upon each other along the line joining them [2][3] The differences between the Weak and Strong laws are illustrated below.

Illustration of the Weak Law of Action and Reaction. The forces exerted by A and B upon each other are equal and opposite, but do not necessarily act along the line joining A and B.
Illustration of the Strong Law of Action and Reaction. The forces exerted by A and B upon each other are equal and opposite, and act along the line joining A and B.

The electrostatic force and the gravity are examples of forces which obey the Strong Law; in fact, most forces do. A rock resting on a table pushes down on the table due to gravity, so the table also pushes back up, and both forces are aligned. However, there exist important exceptions, such as the magnetic force.

Validity of Newton's Laws[edit]

Newton's laws are valid only under certain conditions.

  • In general, the distances with which one works must be much greater than the size of atoms and molecules. If this is untrue, then quantum mechanics must be used in place of classical mechanics.
  • In general, the speeds which one works with must be much less than the speed of light. If this untrue, then relativistic mechanics must be used in place of classical mechanics.
  • Even when the above are true, Newton's laws are only valid with respect to inertial frames of reference.

Broadly speaking, a frame of reference is a standard against which velocity and acceleration and so forth are measured. For example, the speed of a car is usually measured with respect to the surface of the Earth. An inertial frame of reference, in particular, is a frame of reference which is not accelerating (it is not orbiting, rotating, or speeding up). For example, Newton's laws do not apply if the frame of reference considered is an accelerating train; and for that matter, neither is the Earth an inertial frame of reference.

Conversely, it is possible to define an inertial frame of reference as being a frame of reference in which Newton's first and second laws apply.

References[edit]

  1. Weisstein, Eric. "Weak Law of Action and Reaction", Eric Weisstein's World of Physics. Accessed May 13, 2007.[1]
  2. Weisstein, Eric. "Strong Law of Action and Reaction", Eric Weisstein's World of Physics. Accessed May 13, 2007.[2]
  3. Goldstein, Herbert. Classical Mechanics. Reading, Massachusetts: Addison-Wesley, 1950, p. 4.