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.