Definition of Work

Definition of Work in Physics

When a force acts to move an object, we say that Work was done on the object by the Force.

Work done by a Constant Force:

	= Force vector applied to the object/system.
F_s	= Component of Force along the direction of movement.
	= Displacement vector.
s	= Distance the system is displaced.
f	= Angle between the displacement and the force.
	= Scalar or Dot product of the force vector and the distance vector

Work is usually defined in terms of the dot product because it is only the component of the force along the direction of motionof the object - F cos(q) - which does any work and the dot product nicely expresses product in a compact form. One corollary is that the perpendicular component of the force can never do any work on object.
Work is the measure of a quantity that is capable of accomplishing Macroscopic Motion of a System due to the action of a Force over a Distance.
Force is the agent of change, and Work is a measure of the change.
The Force does the Work, not the agent that created the Force. Do not confuse the work you do to create a force with the work done by the force you create; they are not the same. The force you exert holding a 100 pound barbell above your head does no work on the barbell while the barbell is at rest, but you do work (on a molecular level) to create that force.
Work is related to the distance a force moves an object and not the time it takes to move the object. See Work Comparsion QT Movie.
A Force does no work unless the system is free to move "along the direction" of the Force applied. When a Force and the object's displacement are perpendicular, the work done by the force is zero.

Work Done By a Variable Force

If the force is not constant along the path of the object we need to calculate the over very tiny intervals and then add them up. This is exactly what the integration over differential small intervals of link can accomplish.

	= Force vector function that is applied to the body.
F_s(s)	=Component of force function along the direction of movement.
F(s)	= Function of the Magnitue of the force vector along the displacement curve.
	= Dot or Scalr product of the force vector and the differential displacement vector
d	= Differential displacement vector.
ds	= Differential displaced.
f	= Angle between displacement and the force.
s_o	= Initial location of the body
s_f	= Final location of the body

When the force is constant this definition of Work becomes identical to that above for the work done by a constant force.

In order to calculate the work done by a variable force, one needs to know how the force changes over the path of the object's motion.

Example: Work done by Spring
The force that a spring exerts on an object attached to its end not constant as the spring is stretched and is given by F(s) = -k s (see Hook's Law). To calculate the work done by the spring when it is stretched from x_o to x_f we integrate F(s) over ds.

The angle between F and ds is 180^o since the force is in the opposite direction of the motion of the end of the spring when the spring is being stretched.

UNITS:

Other Common units of Work and Energy

Two Examples of Work	Force	External Work Needed	Work done by the Force
Gravity (near the Earth's surface)	F_grav = –mg	W_ext = mg(h₂ - h₁)	W_grav = –mg(h₂ - h₁)
Spring (Hooken Spring)	F_spring = –kx	W_ext = 1/2 k (x₂² - x₁²)	W_spring =–1/2 k (x₂² - x₁²)

Work Comparison
The work done on three blocks of different mass are compared when the same size force acts over the same distance. Displayed is the transformation of work into kinetic energy of the blocks. Work Comparsion QT Movie (240K)

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