## Work = Force times Distance = Energy

*Work* is defined as *force times distance*. Work is a measure
of the *energy* expended in applying a force to move an
object.^{B.8}

The work required to compress a spring through a displacement of meters, starting from rest, is then

Work can also be negative. For example, when uncompressing an
ideal spring, the (positive) work done by the spring on its moving end
support can be interpreted also as saying that the end support
performs negative work on the spring as it allows the spring to
uncompress. When negative work is performed, the driving system is
always accepting energy from the driven system. This is all simply
accounting. Physically, one normally considers the *driver* as
the agent performing the positive work, *i.e.*, the one expending energy
to move the driven object. Thus, when allowing a spring to
uncompress, we consider the spring as performing (positive) work on
whatever is attached to its moving end.

During a sinusoidal mass-spring oscillation, as derived in §B.1.4, each period of the oscillation can be divided into equal sections during which either the mass performs work on the spring, or vice versa.

Gravity, spring forces, and electrostatic forces are examples of
*conservative forces*. Conservative forces have the property
that the work required to move an object from point to point ,
either with or against the force, depends only on the locations of
points and in space, not on the path taken from to .

### Potential Energy in a Spring

When compressing an ideal spring, *work* is performed, and this
work is *stored* in the spring in the form of what we call
*potential energy*. Equation (B.6) above gives the quantitative
formula for the potential energy stored in an ideal spring
after it has been compressed meters from rest.

### Kinetic Energy of a Mass

*Kinetic energy* is energy associated with *motion*. For
example, when a spring uncompresses and accelerates a mass, as in the
configuration of Fig.B.2, work is performed on the mass
by the spring, and we say that the potential energy of the spring is
converted to *kinetic energy* of the mass.

Suppose in Fig.B.2 we have an initial spring compression by meters at time , and the mass velocity is zero at . Then from the equation of motion Eq.(B.5), we can calculate when the spring returns to rest (). This first happens at the first zero of , which is time . At this time, the velocity, given by the time-derivative of Eq.(B.5),

### Mass Kinetic Energy from Virtual Work

From Newton's second law,
(introduced in Eq.(B.1)),
we can use d'Alembert's idea of *virtual work* to derive the
formula for the kinetic energy of a mass given its speed
.
Let denote a small (infinitesimal) displacement of the mass in
the direction. Then we have, using the calculus of differentials,

Thus, by Newton's second law, a differential of work applied to a mass by force through distance boosts the kinetic energy of the mass by . The kinetic energy of a mass moving at speed is then given by the integral of all such differential boosts from 0 to :

*kinetic energy*of mass traveling at speed .

The quantity is classically called the *virtual work*
associated with force , and a *virtual displacement*
[544].

### Energy in the Mass-Spring Oscillator

Summarizing the previous sections, we say that a compressed spring
holds a *potential energy* equal to the *work* required to
compress the spring from rest to its current displacement. If a
compressed spring is allowed to expand by pushing a mass, as in the
system of Fig.B.2, the potential energy in the spring
is converted to *kinetic energy* in the moving mass.

We can draw some inferences from the oscillatory motion of the mass-spring system written in Eq.(B.5):

- From a global point of view, we see that
*energy is conserved*, since the oscillation never decays. - At the
*peaks*of the displacement (when is either or ), all energy is in the form of*potential*energy,*i.e.*, the spring is either maximally compressed or stretched, and the mass is momentarily stopped as it is changing direction. - At the
*zero-crossings*of , the spring is momentarily relaxed, thereby holding no potential energy; at these instants, all energy is in the form of*kinetic*energy, stored in the motion of the mass. - Since total energy is conserved (§B.2.5), the kinetic
energy of the mass at the displacement zero-crossings is exactly the
amount needed to stretch the spring to displacement (or compress
it to ) before the mass stops and changes direction. At all
times, the total energy is equal to the sum of the potential
energy stored in the spring, and the kinetic energy
stored in the mass:

### Energy Conservation

It is a remarkable property of our universe that *energy is
conserved* under all circumstances. There are no known exceptions to
the conservation of energy, even when relativistic and quantum effects
are considered.^{B.9}

*Energy* may be defined as the ability to do
*work*, where work may be defined as *force times distance*
(§B.2).

### Energy Conservation in the Mass-Spring System

Recall that Newton's second law applied to a mass-spring system, as in §B.1.4, yields

Thus, Newton's second law and Hooke's law imply conservation of energy in the mass-spring system of Fig.B.2.

**Next Section:**

Momentum

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Newton's Laws of Motion