Download Work Energy concepts

KEY CONCEPTS
WORK, ENERGY, AND POWER
1. WORK AND THE WORK ENERGY THEOREM
The concept of work is used to describe what is accomplished when a force acts
upon an object as it moves through some displacement. Consider an object moving in a
straight line under the influence of a force. If the force is parallel to the displacement,
then the work done is the product of the magnitudes of the force and the displacement:
W = F|| d
Note that work is a scalar quantity. In practice, force and the resulting displacement are
not always in the same direction. If the angle between the force and the direction is θ,
then a more general equation describing the work performed is:
W = Fd cos θ
In vector notation this is written as:
W = F•d
Notice that with larger angles between the force and the displacement, greater force is
required to achieve the same amount of work. When the force and displacement are in
roughly opposite directions, the work can be a negative value. Furthermore, if an object
does not move, no matter how great a force is applied to it the work is always zero. The
unit of work is the Joule (J) and is equivalent to one N⋅ m.
Using Newton’s second law, an equation can be derived relating the net work to
the mass and velocity of the object and ultimately to its kinetic energy:
mv 2 mv02
−
= K − K0 = ΔK
Wnet =
2
2
where K is the kinetic energy of an object, K0 is its initial kinetic energy and ΔK is the
change in its kinetic energy. Thus the net work done on an object is equal to the change
in its kinetic energy. This is known as the work-energy theorem. If the force changes
over the course of the displacement, we must replace the product with an integral, and
the work equation becomes:
KEY CONCEPTS
WORK, ENERGY, AND POWER
W=
∫ Fdx
2. CONSERVATIVE FORCES AND POTENTIAL ENERGY
A conservative force depends only on an object’s position and not on its path.
A non-conservative force may depend on either or both. As a result, if an object
under the influence of a conservative force returns to its original position, it also regains
to its original kinetic energy. The same cannot be said of an object subjected to a nonconservative force. Gravity is a conservative force, whereas friction is a nonconservative force.
When an object is acted upon by conservative forces it can have a potential
energy, U, associated with those forces. Therefore, work also can be defined in terms of
the change in potential energy as:
ΔU = −W
In the case of the force due to gravity, the change in potential energy is:
ΔU g = −W = mgΔy = mgh
where h is the change in vertical height. The zero point of potential energy is arbitrary,
so for convenience it can be set at y = 0. The gravitational potential energy thus
becomes:
U g = mgy = mgh
If the force changes with position, such as with a spring:
Fs = −kx
then the potential energy can be calculated using:
x
U s( x ) = − ∫ F( x )dx + U 0
0
where U0 is the potential energy at x=0. If we set the zero point of the potential energy
U0 to zero, then the result is:
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KEY CONCEPTS
WORK, ENERGY, AND POWER
Us =
kx 2
2
3. CONSERVATION OF ENERGY
The work done by non-conservative forces is equal to the total change in kinetic
and potential energy as given by:
ΣWnc = ΔK + ΔU
In the special case where no non-conservative forces are acting upon an object, Wnc = 0
and the above equation becomes:
ΔK + ΔU = Δ ( K + U ) = 0
In other words, the total of the kinetic and potential energy does not change. Thus, the
total mechanical energy, E, remains constant and is given by:
E = K + U = constant
This is the law of conservation of mechanical energy. In the real world, nonconservative forces such as friction and air resistance act upon moving objects. These
forces convert some of the energy to thermal energy, which is sometimes referred to as
its internal energy, EI. Therefore, a more general statement regarding the law of
conservation of energy can be made as:
E = K + U + EI = constant
In this case E is no longer the mechanical energy, but the total energy of the system.
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KEY CONCEPTS
WORK, ENERGY, AND POWER
4. POWER
Power is the rate at which work is done. The average power is equal to the work
divided by the elapsed time or:
Pavg =
W
t
If the power is constant, the instantaneous power equals the average power. The work
associated with this constant power is given by:
W = Pt
If the power changes with time, then the equation for power becomes:
P=
dW
dt
and the equation for work becomes:
W=
∫ Pdt
The unit of power is the Watt (W). One watt is equal to one J/s. In other words, if you
do one joule of work in every second you have a power of 1 W. Power can also be
related to the velocity by the equation:
P=
F •d
d
=F• =F•v
t
t
This equation is valid even when the force and velocity are changing, or when they are
not parallel to each other.
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