### PHY1025F-2014-M04-Energy-Lecture Slides

```Physics 1025F
Mechanics
ENERGY
Dr. Steve Peterson
[email protected]
UCT PHY1025F: Mechanics
1
Chapter 6: Work and Energy
We have been using forces to study the
translational motion of objects; Energy (and
work) can provide an alternate analysis of this
motion
UCT PHY1025F: Mechanics
2
ENERGY
Energy …
is an extremely abstract concept and is difficult to define;
is a number (a scalar) describing the state of a system of objects (for
an isolated system this number remains constant, i.e. the energy of
the system is conserved);
appears in many different forms, each of which can be converted into
another form of energy in one or other of the transformation
processes which underlie all activity in the Universe;
is all there is! (Even matter is energy: E = mc2)
UCT PHY1025F: Mechanics
3
3
Systems and Energy
Although energy is hard to define and comes in many
different forms, every system in nature has associated with
it a quantity we call its total energy.
The total energy (E) is the sum of all the different forms of
energy present in the system, i.e.
E  K E  U G  U S  E th  E chem  ...
Energy transformations can occur within a system.
UCT PHY1025F: Mechanics
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System & Energy Transformation
A system is what we define it to be.
Energy can be transformed within the system without
loss.
Energy is a property of a system.
UCT PHY1025F: Mechanics
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Environment & Energy Transfers
An exchange of energy between system and environment is
called an energy transfer.
Two primary energy-transfer processes: Work & Heat
Work is a mechanical transfer of energy to or from a system
by pushing or pulling it.
Heat is a non-mechanical transfer of energy from the
environment to the system (or vice versa) because of a
temperature difference between the two.
UCT PHY1025F: Mechanics
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Work-Energy Principle
Work done on a system represents energy that is
transferred into or out of the system.
The energy of the system (ΔE) changes by the exact amount
of work (W) that was done.
E  W
Work-Energy Principle: The total energy of the system
changes by the amount of work done on it.
 E   KE   U G   U S   E th  ...  W
UCT PHY1025F: Mechanics
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Conservation of Energy
Suppose we have an isolated system, separating it from its
surroundings in such a way that no energy is transferred
into or out of the system.
E  W  0
Law of Conservation of Energy: The total energy of an
isolated system remains constant.
E  constant
UCT PHY1025F: Mechanics
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Feynman on Energy
There is a fact, or if you wish, a law, governing all natural
phenomena that are known to date. There is no known
exception to this law—it is exact so far as we know. The law is
called the conservation of energy. It states that there is a certain
quantity, which we call energy, that does not change in the
manifold changes which nature undergoes. That is a most
abstract idea, because it is a mathematical principle; it says that
there is a numerical quantity which does not change when
something happens. It is not a description of a mechanism, or
anything concrete; it is just a strange fact that we can calculate
some number and when we finish watching nature go through
her tricks and calculate the number again, it is the same.
- Richard Feynman
UCT PHY1025F: Mechanics
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How to Calculate Work
The work done by a constant force F on an object is equal
to the product of the force multiplied by the distance
through which the force acts.
 
W  F d
Dot Product: Vector
Multiplication
Therefore if the motion is in the same direction as the
applied force the magnitude of the work done W is:
W  F d
UCT PHY1025F: Mechanics
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How to Calculate Work
If, on the other hand, the applied force F makes an angle θ
with the subsequent displacement, d then the work done is
W  F d   F cos   d

F


d
W  Fd cos 
Note: Work is a scalar quantity
UCT PHY1025F: Mechanics
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Work can be positive or negative
- Positive if the force and the displacement
are in the same direction (θ = 0°)
- Negative if the force and the displacement
are in the opposite direction (θ = 180°)
Work can also be zero
- If the displacement is perpendicular to the
force (θ = 90°)
UCT PHY1025F: Mechanics
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Units of Work
In the SI system, the units of work are joules:
1 J 1 N m 1
kg  m
s
UCT PHY1025F: Mechanics
2
2
13
More on Work
Work is positive when lifting the
box
Work would be negative if lowering
the box
- The force would still be upward,
but the displacement would be
downward
UCT PHY1025F: Mechanics
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Example: Work
A sled loaded with bricks has a total mass of 18.0 kg and is
pulled at constant speed by a rope inclined at 20.0° above
the horizontal. The sled moves a distance of 20.0 m on a
horizontal surface. The coefficient of friction between the
sled and surface is 0.500. (a) What is the tension in the
rope? (b) How much work is done by the rope on the sled?
(c) What is the mechanical energy lost due to friction?
UCT PHY1025F: Mechanics
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Kinetic Energy
Kinetic energy is the energy of motion.
All moving objects have kinetic energy.
KE 
1
mv
2
2
UCT PHY1025F: Mechanics
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Potential Energy
It is sometimes possible within a system to store energy so
that it can be easily recoverable.
This sort of stored energy is called potential energy.
We will look at gravitational potential energy (due to the
force of gravity) and elastic potential energy (due to the
force from a spring).
Interaction forces that can store useful energy are called
conservative forces.
UCT PHY1025F: Mechanics
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Gravitational Potential Energy
Gravitational potential energy (UG) depends only on the
height of the object and not the path the objects took to
get to that position.
U G  mgy
Assuming UG = 0 when y = 0
UCT PHY1025F: Mechanics
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Elastic Potential Energy
The force exerted by a spring (FS) is called Hooke’s Law.
F S   kx
Energy can be stored in a spring as elastic potential
energy (US).
US 
UCT PHY1025F: Mechanics
1
kx
2
2
19
Thermal Energy
Thermal energy is related to the microscopic motion of
the molecules of an object.
The molecule’s motion produces kinetic energy and the
spring-like molecular bonds produce potential energy.
The sum of these microscopic kinetic and potential
energies is what we call thermal energy.
UCT PHY1025F: Mechanics
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Work & Thermal Energy
If work is done in the presence of friction, then thermal
energy (heat) is generated - heat is another form of
energy and therefore some of the work has gone into
producing the heat.

F fr

F
W   KE   E th
UCT PHY1025F: Mechanics
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Law of Conservation of Energy
In general,
W   KE   PE   Heat
i.e. the work done on the body is converted into changes in
KE and/or changes in PE and/or changes in heat.
W  ( KE
 KE i )  ( PE
f
W  ( KE
f
 PE
f
f
 PE i )  ( H
f
 Hi)
 H f )  ( KE i  PE i  H i )
W  E f  Ei 
E f  Ei  W
Any change in the energy of a system is the result of work done on the system
UCT PHY1025F: Mechanics
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Law of Conservation of Energy
So, if there is no work done on the system?
E f  Ei  W

E f  Ei
This gives rise to the Law of Conservation of Energy which
can be stated as:
"Energy can be neither created nor destroyed, but
can be converted from one form to another or transferred
from one system to another”.
UCT PHY1025F: Mechanics
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Mechanical Energy
If there is no friction present and no external forces (other
than gravity) acting on the system we have
 KE   PE  0
or
KE
f
 PE
f
 KE i  PE i
This is a very powerful equation, and we often refer to the
sum of KE and PE as "mechanical energy”.
UCT PHY1025F: Mechanics
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Conservation of Mechanical Energy
What is conservation in Physics?
- To say a physical quantity is conserved is to say that the
numerical value of the quantity remains constant
throughout any physical process although the quantities
may change form.
In Conservation of Energy, the total mechanical energy
remains constant
- In any isolated system of objects interacting only through
conservative forces, the total mechanical energy of the
system remains constant.
UCT PHY1025F: Mechanics
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Conservative & Nonconservative Forces
There are two general kinds of forces
• Conservative
– Work and energy associated with the force can be recovered
• Nonconservative
– The forces are generally dissipative and work done against it
cannot easily be recovered
Potential energy can only be
forces.
UCT PHY1025F: Mechanics
defined for conservative
26
Example: Energy Conservation
A stone is dropped from a 60-m high cliff onto the ground
below. (a) What is the speed of the stone when it hits the
ground?
(b) Now, the stone is thrown upwards at 20 m/s from the
top of the cliff. What is the speed of the stone when it hits
the ground?
(c) How would the final speed change if the stone were
thrown upward at an angle?
UCT PHY1025F: Mechanics
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How Quickly is Energy Transformed?
The rate at which energy is transformed is called the power
(P) and defined as:
P 
E
t

P 
W
t
Power is also defined as the rate at which work is done.
In the SI system, the units of power are measured in joules
per second or watts (W):
1W  1
J
s
UCT PHY1025F: Mechanics
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Example: Energy Conservation
A 2-kg block is pulled up a frictionless incline (30° above
horizontal) by a 15 N force. What is the speed of the block
after traveling 6-m?
UCT PHY1025F: Mechanics
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Example: Energy Conservation
1-kg and 2-kg masses hang from opposite ends of a string
hanging over a frictionless pulley. The 1-kg mass sits on the
ground and the 2-kg mass is 5-m in the air. With what
speed will the 2-kg mass hit the ground?
UCT PHY1025F: Mechanics
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