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5.3. Conservation of Energy
Conservation of Energy
Energy is never created or destroyed. Any
time work is done, it is only transformed
from one form to another:
• Kinetic Energy
• Potential Energy
– Gravitational, elastic
– Electromagnetic, chemical potential (bonds)
• Thermal Energy
– The kinetic energy of molecules
Mechanical Energy
Mechanical Energy is the sum of potential
and kinetic energy: ME = PE + KE.
In a system without friction (= loss to thermal
energy), mechanical energy is conserved
i.e. constant.
Conservation of mechanical energy (or
mechanical + thermal energy) can be used
to solve various problems.
• Pendulum demo
Note PE can be 𝑚𝑔ℎ, 1
2𝑘 ∆𝑥 2 or both!
Example
You drop a 1.0-kg stone from a height if 2.0
m. What is its velocity when it hits the
ground?
Solution 1: Equations of Motion
𝑣𝑓2 = 𝑣𝑖
2 + 2𝑎∆𝑥 = 0 + 2𝑔ℎ
𝑣𝑓 = 2𝑔ℎ
Solution 2: Conservation of Energy
𝑃𝐸𝑖 + 𝐾𝐸𝑖 = 𝑃𝐸𝑓 + 𝐾𝐸𝑓
𝑚𝑔ℎ + 0 = 0 + 12𝑚𝑣2
𝑣2 = 2𝑔ℎ → 𝑣 = 2𝑔ℎ
Example 2
A batter hits a baseball, which leaves the bat
at 36 m/s. A fan in the outfield bleachers,
7.2 m above, catches it. What was its speed
when caught?
𝑃𝐸𝑖 + 𝐾𝐸𝑖 = 𝑃𝐸𝑓 + 𝐾𝐸𝑓
0 + 12𝑚𝑣𝑖
2 = 𝑚𝑔ℎ + 12𝑚𝑣𝑓
2
𝑣𝑓2 = 𝑣𝑖
2 − 2𝑔ℎ
𝑣 = 36 2 − 2 9.81 7.2 = 34 m/s
Example 3
If friction is negligible, which swimmer has
greater speed at the bottom of the slide?
The path did not matter! Gravity is called a
conservative force. The change in PEgrav depends
only on the initial and final positions, not path. This
allows us to solve problems with variable 𝑎.
Conservative and Nonconservative Forces
Conservative force: the work done by or against it
is stored in the form of potential energy that can be
released (recovered) at a later time
Examples of a conservative force: gravity, spring
Nonconservative force: the work by or against it is
“lost”, and not stored as potential energy
Example of a nonconservative force: friction
Work Done by a Conservative Force
Work done by a conservative force (e.g. gravity)
around a closed path is zero.
Work Done by a Conservative Force
The work done by a conservative force does
not depend on the path:
Consequently
the work done or
required can be
computed from
only the initial
and final
positions.
Work Done by a Nonconservative Force
Work done by a nonconservative force (e.g.
friction) around a closed path is not zero:
Non-conservative Forces
The work done by friction is converted from kinetic
energy into thermal energy (which is the KE of the
molecules). Since the frictional force depends on
the direction of motion, it depends on the path
taken. Potential energy [fields] cannot be defined for
non-conservative forces.
Conservative Forces and Work
Which requires more
work, lifting a box
straight up (𝑊1) or
sliding it up a
frictionless ramp at
constant velocity (𝑊2)?
𝑊2 = 𝐹𝑑 = 𝐹2𝐿= 𝑚𝑔 sin𝜙 𝐿= 𝑚𝑔 ℎ 𝐿 𝐿= 𝑚𝑔ℎ = 𝑊1
Why then is a ramp
used?
Potential Energy Curves
The curve of a hill or a roller coaster is itself
essentially a plot of the gravitational
potential energy:
Potential Energy Curves
The potential energy curve for a spring:
Conservation of Energy
Energy can be transferred from one object
to another, e.g. when a compressed spring
transfers its potential energy to the kinetic
energy of a ball. In any case, the total
energy of the two-object system is
conserved (constant).
• Read/try the great examples in the book in
the Conservation of Energy section!
• Thermal, electrical, nuclear and chemical
energy can be described in terms of kinetic
energy and potential energy.
• Nuclear reactions indicate that mass can
be converted to energy (E=mc2), and thus
is a form of energy.
• Loop-the-loop demo: where did the PE go?
Hamilton’s Principle I
𝑚𝑔ℎ + 1
2𝑚𝑣2 = 𝐶 Conservation of Energy
𝑚𝑔𝜕ℎ
𝜕𝑡+
1
2𝑚 2𝑣
𝜕𝑣
𝜕𝑡= 0 Take
𝜕
𝜕𝑡of both sides.
𝑚𝑔𝑣 + 𝑚𝑣𝑎 = 0 Simplify.
𝑚𝑔 + 𝑚𝑎 = 0 Divide by 𝑣.
−𝐹𝑔𝑟𝑎𝑣𝑖𝑡𝑦 + 𝑚𝑎 = 0 Newton’s 2nd Law!
Thus Newton’s 2nd Law (conservation of
momentum) is a manifestation of conservation
of energy.
Hamilton’s Principle II
1
2𝑘𝑥2 + 1
2𝑚𝑣2 = 𝐶 Conservation of Energy
1
2𝑘 2𝑥
𝜕𝑥
𝜕𝑡+
1
2𝑚 2𝑣
𝜕𝑣
𝜕𝑡= 0 Take
𝜕
𝜕𝑡.
𝑘𝑥𝑣 + 𝑚𝑣𝑎 = 0 Simplify.
𝑘𝑥 + 𝑚𝑎 = 0 Divide by 𝑣.
−𝐹𝑠𝑝𝑟𝑖𝑛𝑔 + 𝑚𝑎 = 0 Newton’s 2nd Law!
Thus Newton’s 2nd Law (conservation of
momentum) is a manifestation of conservation
of energy.