Additional examples, video, and practice available at WileyPLUSREVIEW & SUMMARY 199Substituting Eqs. 8-43 through 8-45 into Eq. 8-42, we find
0 kd^2 mghmkmgL, (8-46)
and
L69.3 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.(3.20 103 N/m)(5.00 m)^2
2(0.800)(200 kg)(9.8 m/s^2 )35 m
0.800kd^2
2 mkmgh
mk1
2Conservative Forces A force is a conservative forceif 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 par-
ticle. The gravitational force and the spring force are conservative
forces; the kinetic frictional force is a nonconservative force.
Potential Energy Apotential energyis energy that is associated
with the configuration of a system in which a conservative force acts.
When the conservative force does work Won a particle within the sys-
tem, the change Uin the potential energy of the system is
UW. (8-1)If the particle moves from point xito point xf, the change in the
potential energy of the system is
(8-6)Gravitational Potential Energy The potential energy asso-
ciated with a system consisting of Earth and a nearby particle is
gravitational potential energy.If the particle moves from height yi
to height yf, the change in the gravitational potential energy of the
particle–Earth system is
Umg(yfyi)mgy. (8-7)If the reference pointof the particle is set as yi0 and the cor-
responding gravitational potential energy of the system is set as
Ui0, then the gravitational potential energy Uwhen the parti-
U
xf
xiF(x)dx.Review & Summary
cle is at any height yis
U(y)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 Fkxwhen
its free end has displacement x, the elastic potential energy is
(8-11)
The reference configurationhas the spring at its relaxed length, at
whichx0 and U0.Mechanical Energy The mechanical energyEmecof a system
is the sum of its kinetic energy Kand potential energy U:
EmecKU. (8-12)
Anisolated systemis one in which no external forcecauses energy
changes. If only conservative forces do work within an isolated sys-
tem, then the mechanical energy Emecof the system cannot change.
This principle of conservation of mechanical energyis written as
K 2 U 2 K 1 U 1 , (8-17)
in which the subscripts refer to different instants during an
energy transfer process. This conservation principle can also be
written as
EmecKU0. (8-18)Potential Energy Curves If we know the potential energy
functionU(x) for a system in which a one-dimensional force F(x)U(x)^12 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
Emec,2 K 2 Ue 2 Ug 2
0 0 0. (8-44)Let’s next go after the change Ethof the thermal energy of
the glider and ground-level track. From Eq. 8-31, we can
substitute for EthwithfkL(the product of the frictional
force magnitude and the distance of rubbing). From Eq. 6-2,
we know that fk mkFN, where FNis the normal force.
Because the glider moves horizontally through the region
with friction, the magnitude of FNis equal to mg(the up-
ward force matches the downward force). So, the friction’s
theft from the mechanical energy amounts to
Eth mkmgL. (8-45)(By the way, without further experiments, we cannotsay
how much of this thermal energy ends up in the glider and
how much in the track. We simply know the total amount.)