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199
REVIEW & SUM MARY
Substituting Eqs. 8-43 through 8-45 into Eq. 8-42, we find
0 ϭ kd
2 ϩ mgh Ϫ m k mgL,
( 8 - 4 6 )
and
L
ϭ 69.3 m.
(Answer)
Finally, note how algebraically simple our solution is. By
carefully defining a system and realizing that we have an
isolated system, we get to use the law of the conservation of
energy. That means we can relate the initial and final states
of the system with no consideration of the intermediate
states. In particular, we did not need to consider the glider as
it slides over the uneven track. If we had, instead, applied
Newton’s second law to the motion, we would have had to
know the details of the track and would have faced a far
more difficult calculation.
ϭ
(3.20 ϫ 10
3
N/m)(5.00 m)
2
2(0.800)(200 kg)(9.8 m/s
2
)
ϩ
35 m
0.800
ϭ
kd
2
2m k mg
ϩ
h
m k
1
2
Conservative Forces A force is a conservative force if the net
work it does on a particle moving around any closed path, from an
initial point and then back to that point, is zero. Equivalently, a
force is conservative if the net work it does on a particle moving
between two points does not depend on the path taken by the particle. The gravitational force and the spring force are conservative
forces; the kinetic frictional force is a nonconservative force.
Potential Energy A potential energy is energy that is associated
with the configuration of a system in which a conservative force acts.
When the conservative force does work W on a particle within the system, the change ⌬U in the potential energy of the system is
⌬U ϭ ϪW.
( 8 - 1 )
If the particle moves from point x i to point x f , the change in the
potential energy of the system is
(8-6)
Gravitational Potential Energy The potential energy associated with a system consisting of Earth and a nearby particle is
gravitational potential energy. If the particle moves from height y i
to height y f , the change in the gravitational potential energy of the
particle–Earth system is
⌬U ϭ mg(y f Ϫ y i ) ϭ mg ⌬y.
( 8 - 7 )
If the reference point of the particle is set as y i ϭ 0 and the corresponding gravitational potential energy of the system is set as
U i ϭ 0, then the gravitational potential energy U when the parti⌬U ϭ Ϫ͵
x f
x i
F(x) dx.
Review & Summary
cle is at any height y is
U(y) ϭ mgy.
( 8 - 9 )
Elastic Potential Energy Elastic potential energy is the
energy associated with the state of compression or extension of an
elastic object. For a spring that exerts a spring force F ϭ Ϫkx when
its free end has displacement x, the elastic potential energy is
(8-11)
The reference configuration has the spring at its relaxed length, at
which x ϭ 0 and U ϭ 0.
Mechanical Energy The mechanical energy E mec of a system
is the sum of its kinetic energy K and potential energy U:
E mec ϭ K ϩ U.
( 8 - 1 2 )
An isolated system is one in which no external force causes energy
changes. If only conservative forces do work within an isolated system, then the mechanical energy E mec of the system cannot change.
This principle of conservation of mechanical energy is written as
K 2 ϩ U 2 ϭ K 1 ϩ U 1 ,
( 8 - 1 7 )
in which the subscripts refer to different instants during an
energy transfer process. This conservation principle can also be
written as
⌬E mec ϭ ⌬K ϩ ⌬U ϭ 0.
(8-18)
Potential Energy Curves If we know the potential energy
function U(x) for a system in which a one-dimensional force F(x)
U(x) ϭ
1
2 kx
2
.
In the final state, with the spring now in its relaxed state and
the glider again stationary but no longer elevated, the final
mechanical energy of the system is
E mec,2 ϭ K 2 ϩ U e2 ϩ U g2
ϭ 0 ϩ 0 ϩ 0.
(8-44)
Let’s next go after the change ⌬E th of the thermal energy of
the glider and ground-level track. From Eq. 8-31, we can
substitute for ⌬E th with f k L (the product of the frictional
force magnitude and the distance of rubbing). From Eq. 6-2,
we know that f k ϭ m k F N , where F N is the normal force.
Because the glider moves horizontally through the region
with friction, the magnitude of F N is equal to mg (the upward force matches the downward force). So, the friction’s
theft from the mechanical energy amounts to
⌬E th ϭ m k mgL.
( 8 - 4 5 )
(By the way, without further experiments, we cannot say
how much of this thermal energy ends up in the glider and
how much in the track. We simply know the total amount.)
199
REVIEW & SUM MARY
Substituting Eqs. 8-43 through 8-45 into Eq. 8-42, we find
0 ϭ kd
2 ϩ mgh Ϫ m k mgL,
( 8 - 4 6 )
and
L
ϭ 69.3 m.
(Answer)
Finally, note how algebraically simple our solution is. By
carefully defining a system and realizing that we have an
isolated system, we get to use the law of the conservation of
energy. That means we can relate the initial and final states
of the system with no consideration of the intermediate
states. In particular, we did not need to consider the glider as
it slides over the uneven track. If we had, instead, applied
Newton’s second law to the motion, we would have had to
know the details of the track and would have faced a far
more difficult calculation.
ϭ
(3.20 ϫ 10
3
N/m)(5.00 m)
2
2(0.800)(200 kg)(9.8 m/s
2
)
ϩ
35 m
0.800
ϭ
kd
2
2m k mg
ϩ
h
m k
1
2
Conservative Forces A force is a conservative force if the net
work it does on a particle moving around any closed path, from an
initial point and then back to that point, is zero. Equivalently, a
force is conservative if the net work it does on a particle moving
between two points does not depend on the path taken by the particle. The gravitational force and the spring force are conservative
forces; the kinetic frictional force is a nonconservative force.
Potential Energy A potential energy is energy that is associated
with the configuration of a system in which a conservative force acts.
When the conservative force does work W on a particle within the system, the change ⌬U in the potential energy of the system is
⌬U ϭ ϪW.
( 8 - 1 )
If the particle moves from point x i to point x f , the change in the
potential energy of the system is
(8-6)
Gravitational Potential Energy The potential energy associated with a system consisting of Earth and a nearby particle is
gravitational potential energy. If the particle moves from height y i
to height y f , the change in the gravitational potential energy of the
particle–Earth system is
⌬U ϭ mg(y f Ϫ y i ) ϭ mg ⌬y.
( 8 - 7 )
If the reference point of the particle is set as y i ϭ 0 and the corresponding gravitational potential energy of the system is set as
U i ϭ 0, then the gravitational potential energy U when the parti⌬U ϭ Ϫ͵
x f
x i
F(x) dx.
Review & Summary
cle is at any height y is
U(y) ϭ mgy.
( 8 - 9 )
Elastic Potential Energy Elastic potential energy is the
energy associated with the state of compression or extension of an
elastic object. For a spring that exerts a spring force F ϭ Ϫkx when
its free end has displacement x, the elastic potential energy is
(8-11)
The reference configuration has the spring at its relaxed length, at
which x ϭ 0 and U ϭ 0.
Mechanical Energy The mechanical energy E mec of a system
is the sum of its kinetic energy K and potential energy U:
E mec ϭ K ϩ U.
( 8 - 1 2 )
An isolated system is one in which no external force causes energy
changes. If only conservative forces do work within an isolated system, then the mechanical energy E mec of the system cannot change.
This principle of conservation of mechanical energy is written as
K 2 ϩ U 2 ϭ K 1 ϩ U 1 ,
( 8 - 1 7 )
in which the subscripts refer to different instants during an
energy transfer process. This conservation principle can also be
written as
⌬E mec ϭ ⌬K ϩ ⌬U ϭ 0.
(8-18)
Potential Energy Curves If we know the potential energy
function U(x) for a system in which a one-dimensional force F(x)
U(x) ϭ
1
2 kx
2
.
In the final state, with the spring now in its relaxed state and
the glider again stationary but no longer elevated, the final
mechanical energy of the system is
E mec,2 ϭ K 2 ϩ U e2 ϩ U g2
ϭ 0 ϩ 0 ϩ 0.
(8-44)
Let’s next go after the change ⌬E th of the thermal energy of
the glider and ground-level track. From Eq. 8-31, we can
substitute for ⌬E th with f k L (the product of the frictional
force magnitude and the distance of rubbing). From Eq. 6-2,
we know that f k ϭ m k F N , where F N is the normal force.
Because the glider moves horizontally through the region
with friction, the magnitude of F N is equal to mg (the upward force matches the downward force). So, the friction’s
theft from the mechanical energy amounts to
⌬E th ϭ m k mgL.
( 8 - 4 5 )
(By the way, without further experiments, we cannot say
how much of this thermal energy ends up in the glider and
how much in the track. We simply know the total amount.)