General Mechanics/Work and Power

Work

When a force is exerted on an object, energy is transferred to the object. The amount of energy transferred is called the work done on the object. Mathematically work done is defined as the dot\scalar product of force and displacement, thus it is a scalar quantity. However, energy is only transferred if the object moves. Work can be thought of as the process of transforming energy from one form into another. The work W done is

$W=F\Delta x$

where the distance moved by the object is Δx and the force exerted on it is F. Notice that work can either be positive or negative. The work is positive if the object being acted upon moves in the same direction as the force, with negative work occurring if the object moves opposite to the force.

This equation assumes that the force remains constant over the full displacement or distance( depending on the situation). If it is not, then it is necessary to break up the displacement into a number of smaller displacements, over each of which the force can be assumed to be constant. The total work is then the sum of the works associated with each small displacement. In the infinitesimal limit this becomes an integral

$W=\int F(s)\,ds$

If more than one force acts on an object, the works due to the different forces each add or subtract energy, depending on whether they are positive or negative. The total work is the sum of these individual works.

There are two special cases in which the work done on an object is related to other quantities. If F is the total force acting on the object, then by Newton's second law W=FΔx=mΔx·a . However, a=dv/dt where v is the velocity of the object, and Δx≈vΔt, where Δt is the time required by the object to move through distance Δx . The approximation becomes exact when Δx and Δt become very small. Putting all of this together results in

$W_{total}=m{\frac {dv}{dt}}v\Delta t=\Delta t{\frac {d}{dt}}{\frac {mv^{2}}{2}}$

We call the quantity mv²/2 the kinetic energy, or K. It represents the amount of work stored as motion. We can then say

$W_{total}={\frac {dK}{dt}}\Delta t=\Delta K$

Thus, when F is the only force, the total work on the object equals the change in kinetic energy of the object. This transformation is known as "Work-Energy theorem."

The other special case occurs when the force depends only on position, but is not necessarily the total force acting on the object. In this case we can define a function

$U(x)=-\int ^{x}F(s)\,ds$

and the work done by the force in moving from x₁ to x₂ is U(x₁)-U(x₂), no matter how quickly or slowly the object moved.

If the force is like this it is called conservative and U is called the potential energy. Differentiating the definition gives

$F=-{\frac {dU}{dx}}$

The minus sign in these equations is purely conventional.

If a force is conservative( The force whose effect doesn't depend on the path it has taken to go through), we can write the work done by it as

$W=-{\frac {dU}{dx}}\Delta x=-\Delta U$

where is the change in the potential energy of the object associated with the force of interest.

Energy

The sum of the potential( Energy by virtue of its position) and kinetic(Energy by virtue of its motion) energies is constant. We call this constant the total energy E:

$E=K+U$

If all the forces involved are conservative we can equate this with the previous expression for work to get the following relationship between work, kinetic energy, and potential energy:

$\Delta K=W=-\Delta U$

Following this, we have a very important formula, called the 'Conservation of Energy:

$\Delta (K+U)=0$

This theorem states that the total amount of energy in a system is constant, and that energy can neither be created nor destroyed.

Power

The power associated with a force is simply the amount of work done by the force divided by the time interval over which it is done. It is therefore the energy per unit time transferred to the object by the force of interest. From above we see that the power is

$P={\frac {F\Delta x}{\Delta t}}=Fv$

where $v$ is the velocity at which the object is moving. The total power is just the sum of the powers associated with each force. It equals the time rate of change of kinetic energy of the object:

$P_{total}={\frac {W_{total}}{\Delta t}}={\frac {dK}{dt}}$