Download Notes on Energy

CHAPTER 6 – ENERGY
Because of the nature of the questions asked by the class,
it will not be necessary to cover this chapter in its entirety.
Rather, selected sections will be presented briefly.
Energy is one of the four quantities (along with space,
time and matter) that make up the “physical world”.
Space, time and matter are defined in terms of
fundamental properties: length (in meters), time (in
seconds) and mass (in kilograms). Energy is a derived
quantity, whose units are Joules
1 Joule = 1 kg-m2/s2
= 1 (kg-m/s2)m
= 1 N-m
The last representation above suggests that energy can be
associated with a force which acts over a certain distance
or displacement. (Think of pushing a crate across a floor
50 m to the right by applying a force to the right on the
crate.)
Doing this might make you feel physically tired, i.e. like
you had done a lot of “work”!
So, in physics, work is defined as the product of force (in
the direction of motion) times the distance covered (i.e. the
magnitude of the displacement), or
Work W = Fd (in N-m).
Since the units here are the same as for energy (above), it
follows that “work” and “energy” are essentially the
same. Specifically, we say that energy is the ability (or the
capacity) to do work.
Note: Energy and Work are scalars, i.e. they have no
direction!
In everyday terms, we sometimes talk about many
different kinds of energy: mechanical, thermal, chemical,
electrical, electromagnetic, nuclear. (See Table 6.1,
p.244.)
In reality, however, there are only two distinct kinds of
energy:
1. Kinetic energy – energy due to motion; and
2. Potential energy – energy due to position or shape
Furthermore, the total energy within any system remains
constant. (This is called the Law of Conservation of
Energy).
This means that every physical process can be viewed as a
series of exchanges of energy. When there is more than
one body in the system, energy lost by one (or more) of the
bodies must be gained by the remaining bodies. Even
when there only a single body, an increase (or decrease) in
that body’s potential energy must be accounted for by a
loss (or gain) in its kinetic energy, and vice versa.
Kinetic Energy
As we saw early in the course, the kinetic energy possessed
by a moving body is given the formula
1 2
Kinetic Energy (KE) K  2 mv (kg-m2/s2).
(Note that an object at rest will have KE = 0 J.)
Because energy is a scalar, the v here represents the
magnitude of the velocity vector, i.e. the speed.
(Don’t confuse KE with linear momentum p = mv, which
also depends on mass and velocity, but which is a vector!)
Potential Energy
Potential energy (energy due to position or shape) actually
comes in a number of types: local gravitational potential
energy, universal gravitational potential energy, spring
potential energy, electrostatic potential energy, etc.
The only one we will discuss in this course is the simplest
of these, namely local gravitational potential energy.
Unlike kinetic energy, which is zero when an object is at
rest, potential energy can be assumed to be zero at any
location we choose. What matters is the amount by which
potential energy changes as the object moves. Such a
change can be either positive or negative, depending on
whether the body gains or loses potential energy.
Since we’ve already seen that work and energy have the
same units, it makes sense that the amount of local
gravitational potential energy gained when an object’s
height increases is the work done against gravity to lift it.
Likewise, the amount of local gravitational potential
energy lost when an object descends is the work done by
gravity as it drops.
Since we know that the local force of gravity is just the
weight mg, this work is given by the formula
W = Fd = mgd .
So, we can define the local gravitational potential energy
(relative to any location where we want the energy to be
zero) by
Local Gravitational Potential Energy (PE) = mgy .
Note that, above the zero energy height, y (and hence
PE) will be positive, while below the zero energy height,
 y and PE will be negative.
For instance, suppose we choose PE to be zero on the
surface of a desk which is 1 m high. A 1 kg book would
then have:
PE = 1(9.8)0 = 0
J on the desk (where y = 0 m)
PE = 1(9.8)(+2) = +19.6 J 2 m above the desk
PE = 1(9.8)(–1) = –9.8 J on the floor (1 m below)
Example
A 3 kg stone is released from rest at the top of a 20 m cliff.
As it falls, its velocity and its displacement can be found
using our earlier equations
v f  vi  at  0  9.8t
[1]
1
2
2
y  vi t  at   0  4.9t 
[2] .
2
Find how long it takes for the stone to reach the bottom.
Assuming the local gravitational potential energy (PE) is
zero at the top of the cliff (i.e. at the moment of release),
find the KE, PE and the sum of the two at
the top, at t = 1.0 s, at t = 1.5 s and at the bottom.
Solution
At the bottom, y  20 m, so from [2]
 20
2
t

 2.02 s
 20  4.9t  , i.e.
 4.9
At the top,
t = 0 s and y  0 m, so:
1
2
v f  9.80  0 m/s, i.e. KE = 30  0 J.
2
PE = mgy  39.80  0 J.
It follows that KE + PE = 0 J.
At 1.0 s:
v f  9.81.0  9.8 m/s, i.e. KE =
1
3 9.82  144 J.
2
From [2], y  4.91.0  4.9 m, so
PE = mgy  39.8 4.9  144 J.
2
It follows that KE + PE = 0 J.
At 1.5 s:
1
2



3

14
.
7
 324 J.
v f  9.81.5  14.7 m/s, i.e. KE =
2
From [2], y  4.91.5  11.03 m, so
PE = mgy  39.8 11.03  324 J.
2
It follows that KE + PE = 0 J.
At the bottom, t  2.02 s and y  20 m, so:
1
2
v f  9.82.02  19.8 m/s, i.e. KE = 3 19.8  588 J.
2
PE = mgy  39.8 20  588 J.
It follows that KE + PE = 0 J .
Location/Time
Top (0 s)
1.0 s
1.5 s
Bottom (2.02 s)
Summary
KE
PE
0J
0J
144 J
–144 J
324 J
–324 J
588 J
–588 J
Total
0J
0J
0J
0J
This example clearly illustrates the principle of
conservation of energy: as the rock falls, it accelerates and
gains kinetic energy, but at the same time it loses exactly
the same amount of potential energy, thus keeping the
total energy (0 J, in this particular case) constant
throughout!
Recall that other forms of energy (thermal, chemical,
electrical, nuclear, etc), when considered on a microscopic
scale, are really explained in terms of kinetic and potential
energy. See Table 6.1 on page 244.
For example, everyone knows that adding heat to
substance (solid, liquid or gas) raises its temperature. In
fact, temperature is essentially a measure of the average
kinetic energy of the molecules. Think of gas in a closed
container. Its molecules move very rapidly, continually
colliding with the walls of the container (and sometimes
colliding with each other). What happens to the container
if you heat the gas? If you cool the gas? What would
happen theoretically if the temperature were lowered to
“absolute zero” (0 Kelvin)?
Now that we have gained a basic understanding of kinetic
and potential energy and conservation of total energy, we
are in a position to answer a couple of your questions.