Classical mechanics

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Classical mechanics refers to a branch of physics dealing with every-day laws of mechanics (as opposed to quantum mechanics). The term is sometimes interchangeable with "Newtonian physics", as it was Sir Isaac Newton who first proposed the first laws of motion. However, there are other formulations of classical mechanics that are not due to Newton, namely Lagrangian dynamics and Hamiltonian dynamics. Classical mechanics is sufficient for explaining most observable phenomena.

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Kinematics

Kinematics is the subfield of mechanics that describes motion, without taking into account its cause. The relevant concepts in kinematics are position, velocity and acceleration. The position of an object is usually described by a vector, with respect to some reference point. That position is described as a vector means that we need several numbers to specify it. For example, we can say that object A is three meters to the right and four meters to the front from object B. Note that a single number is not enough to specify its position. We could say that object A is 5 meters from object B, but in this case we need to specify its direction, so, in any case, we need two numbers. Velocity is measures how fast position changes. As such, it is the change in position with respect to time. Velocity is also a vector. Mathematically, it is the derivative of position with respect to time:

v=\frac{dx}{dt}

If the velocity is constant, the distance traveled in a time t is:

x = vt.

Similarly, acceleration measures how fast the velocity changes. It is then mathematically defined as the derivative of velocity with respect to time:

a=\frac{dv}{dt}

If the acceleration is constant, the velocity at a time t is:

v = v0 + at.

Where v0 is the velocity at time t = 0. Integrating this equation gives the distance traveled by an object at constant acceleration:

\Delta x=v_0t+\frac{1}{2}at^2.

Combining previous equations we can find a very useful relation between initial and final velocity and distance traveled, without making reference to time:

v_f^2-v_0^2 = 2a\Delta x

Dynamics

Dynamics is the study of the origin, or causes, of motion. The fundamental concept of dynamics is that of force. Classical dynamics is basically formulated in terms of the three Newton’s laws of motion. Newton’s laws establish the mechanical world view that forms the basis for the scientific revolution of the 17th century. Newton’s three laws are:


1. An object subject to no net external force moves with constant velocity relative to an inertial reference frame. An inertial reference frame is defined as one where Newton's laws are valid. Intuitively, an inertial reference frame is a frame that moves in a straight line at a constant velocity. Since the Earth is rotating, it is not a perfect reference frame; however, it is a very good approximation for everyday purposes.


2. In an inertial reference frame, the acceleration of of an object is equal to the force applied on it, divided by the object mass. Mathematically:

\Sigma \vec F = m \vec a where

  • \Sigma \vec F = total force acting on the object
  • m = mass of the object
  • \vec a = acceleration of the object

This law also applies to a system of objects.

At first sight it seems that Newton’s first law is redundant, in the sense that it is a special case of the second law: If the force is zero, acceleration is zero, and hence the object moves at a constant velocity. However, the two laws are not redundant. The purpose of Newton’s first law is to define what an inertial reference frame is. Once defined, we can formulate Newton’s second law which is only valid on inertial reference frame.

Newton originally wrote this as

\Sigma \vec F = \frac{d \vec p}{dt}

where \vec p is the momentum of the object. Momentum is defined as mass times velocity, p = m*v. This formulation is more general. It reduces to F = m*a when the object has a constant mass.


3. For every action, there is an equal and opposite reaction.

If object A exerts a force on object B, object B will exert a force equal in magnitude and opposite in direction on object A. For example, ff the earth pulls you down with a force of 1500 Newtons, you pull up on the earth with a force of 1500 Newtons. (Of course, since F=ma, and the earth's mass is much greater than yours, the earth accelerates much less than you do.)

Angular Kinematics

These same basic laws are true with respect to angular motion, that is, for a particle moving in a circle. In this case, the position is described by an angle, and is measured in radians. Its first derivative, ω (measured in radians/second), is called angular velocity; its second derivative α is called angular acceleration. The kinematic equations of rotational motion are analogous to those of linear motion:

Δθ = ωt
Δω = ω0 + αt
\Delta \theta = \omega _0 t + \frac{1}{2} \alpha t^2
\omega_f^2-\omega_0^2 = 2 \alpha \Delta \theta

Limitations

Classical mechanics only breaks down in extreme conditions, such as traveling close to the speed of light, being near extremely high gravity, or dealing with subatomic particles. For example, special relativity gives the correction factor at high speeds as

γ=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}

where v is the velocity of an object and c is the speed of light. At 1000 miles per hour, the correction is .999999999999991.

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