Potential Energy and Conservative Forces
Answers to Even-numbered Conceptual Questions
As water vapor rises, there is an increase in the gravitational potential energy of the
Part of this potential energy is released as snow falls onto the mountain.
avalanche occurs, the snow on the mountain accelerates down slope, converting more
gravitational potential energy to kinetic energy.
As the ball falls, gravitational potential energy is converted to kinetic energy. When the
ball hits the floor, some of the kinetic energy is converted to sound energy and heat, some
to a compression of the ball – like compressing a spring.
The ball now rebounds,
converting the potential energy of compression back to kinetic energy.
Finally, the kinetic
energy of the ball is converted back to gravitational potential energy as the ball rises.
final height of the ball is less than its initial height because some energy has left the
system in the form of sound and heat.
The work done in stretching the spring through a doubled distance is the force times the
Both of these quantities increase by a factor of two, and therefore the potential
energy of the spring increases by a factor of four.
We arrive at the same from the form of
the spring potential energy,
, which depends on the square of the amount of
The initial mechanical energy of the system is the gravitational potential energy of the
mass-Earth system. As the mass moves downward, the gravitational potential energy of
the system decreases.
At the same time, the potential energy of the spring increases, as it
is compressed. Initially, the decrease in gravitational potential energy is greater than the
increase in spring potential energy, which means that the mass gains kinetic energy.
Eventually, the increase in spring energy equals the decrease in gravitational energy and
the mass comes to rest.
If a spring is permanently deformed, it will not return to its original length.
As a result,
the work that was done to stretch the spring is not fully recovered – some of it goes into
the energy of deformation.
For this reason, the spring force is not conservative during the
deformation. If the spring is now stretched or compressed by a small amount about its
new equilibrium position, its force is again conservative – though the force constant will
The object’s kinetic energy is a maximum when it is released, and a minimum when it
reaches its greatest height.
The gravitational potential of the system is a minimum
when the object is released, and a maximum when the object reaches its greatest height.
When the term “energy conservation” is used in everyday language, it doesn’t refer to the
total amount of energy in the universe. Instead, it refers to using energy wisely, especially
when a particular source of energy – like oil or natural gas – is finite and nonrenewable.