Transcript Physics 350 - Los Rios Community College District

Chapter 6
Work and Energy
What is Energy?

Transferred into/from different forms
 Heat
 Light
 Sound
 Electrical (batteries)
 Mechanical
 Chemical
○ Food – Calories!

Physics definition:
 “ability to do work”
○ Doesn’t apply to all forms but will apply to mechanical
Introduction

Forms of energy:
 Mechanical 
○
focus for now
 chemical
 electromagnetic
 nuclear

Energy can be transformed from one form to
another
 Essential to the study of physics, chemistry, biology, geology,
astronomy

Can be used in place of Newton’s laws to solve
certain problems more simply (no vectors!)
Work
Provides a link between force and energy
 The work, W, done by a constant force on an
object is defined as the product of the
component of the force along the direction of
displacement and the magnitude of the
displacement

W  (F cos ) x


(F cos θ) is the component of
the force in the direction of the
displacement
Δx is the displacement
Work
 Work
in 1D (along the line of force)
W = Fd




Work Done = Force (on object)
x Displacement
Force acting on a particle moving
along x does an amount of Work
When multiple forces are present,
sum the individual contributions
Work is energy transferred to or from
a system via a force
Work

If the force is not aligned with the x-axis,
we take the x-component

W = (Fcosθ)d
○ F is the magnitude
of the force
○ Δ x is the magnitude
of the object’s displacement
Work

This gives no information about
 the time it took for the displacement to occur
 the velocity or acceleration of the object

Note: work is zero when
○ there is no displacement (holding a bucket)
○ force and displacement are perpendicular
to each other, as cos 90° = 0 (if we are
carrying the bucket horizontally, gravity
does no work)
W  (F cos ) x
(different from everyday “definition” of work)
More About Work

Scalar quantity
Units of Work
SI
joule (J=N m)
CGS
erg (erg=dyne cm)
US Customary foot-pound (foot-pound=ft lb)

If there are multiple forces acting on an object, the
total work done is the algebraic sum of the amount
of work done by each force
More About Work

Work can be positive or negative
 Positive if the force and the displacement are in the
same direction
 Negative if the force and the displacement are in the
opposite direction


Example 1: lifting a cement block…
 Work done by the person:
 is positive when lifting the box
 is negative when lowering the box
 Work done by gravity:
 is negative when lifting the box
 is positive when lowering the box
Example 2: … then moving it horizontally
 is zero when moving it horizontally
T o ta l w o rk :W  W 1  W 2  W 3   m g h  m g h  0  0
lifting lowering moving total
Problem: cleaning the dorm room
Eric decided to clean his dorm room
with his vacuum cleaner. While doing
so, he pulls the canister of the vacuum
cleaner with a force of magnitude
F=50.0 N at an angle 30.0°. He moves
the vacuum cleaner a distance of 3.00
meters. Calculate the work done by all
the forces acting on the canister.
Problem: cleaning the dorm room
Given:
angle:
force:
a=30°
F=55.0 N
1. Introduce coordinate frame:
Oy: y is directed up
Ox: x is directed right
Find:
Work WF=?
Wn=?
Wmg=?
2. Note: horizontal
displacement only,
Work: W=(F cos ) s
WF  ( F cos )x  (50.0 N )(cos30.0 )(3.00m)  130N  m  130J
Wn  ( N cos )x  (n)(cos90.0 )(3.00m)  0 J
Wmg  (mgcos )x  (n)(cos(90.0 ))(3.00m)  0 J

No work as force is
perpendicular to the
displacement
Graphical Representation of Work
 Split total displacement (xf-xi)
into many small displacements x
 For each small displacement:
W i  (F cos ) xi
 Thus, total work is:
W to t 
W 
i
i
Fx   xi
i
which is total area under the F(x) curve!
Work


Example 6.8
A 330-kg piano slides 3.6 m down a 28º incline
and is kept from accelerating by a man who is
pushing back on it parallel to the incline (Fig.
6–36). The effective coefficient of kinetic
friction is 0.40. Calculate:
(a) the force exerted by the man,
(b) the work done by the man on the piano,
(c) the work done by the friction force,
(d) the work done by the force of gravity, and
(e) the net work done on the piano.
To Work or Not to Work
Is it possible to do work on an
1) yes
object that remains at rest?
2) no
To Work or Not to Work
Is it possible to do work on an
1) yes
object that remains at rest?
2) no
Work requires that a force acts over a distance. If an
object does not move at all, there is no displacement,
and therefore no work done.
Kinetic Energy
Energy associated with the motion of an
object
 Scalar quantity with the same units as work
 Work is related to kinetic energy

1
2
KE  mv
2
This quantity is called kinetic energy:
Work-Kinetic Energy Theorem


When work is done by a net force on an
object and the only change in the object is its
speed, the work done is equal to the change
in the object’s kinetic energy
W
n et
 KE f  KEi  KE
 Speed will increase if work is positive
 Speed will decrease if work is negative
KE and Work-Energy Theorem
When work is done by a net force on an
object and the only change in the object
is its speed, the work done is equal to
the change in the object’s kinetic energy
W = KEf – KEi = ΔKE
 Speed will increase if
work is positive
 Speed will decrease if
work is negative
Work and Kinetic Energy

An object’s kinetic
energy can also be
thought of as the
amount of work the
moving object could
do in coming to rest
 The moving hammer
has kinetic energy and
can do work on the nail
Potential Energy

Potential energy is associated with the
position of the object within some system
 Potential energy is a property of the system,
not the object
 A system is a collection of objects or particles
interacting via forces or processes that are
internal to the system

Units of Potential Energy are the same as
those of Work and Kinetic Energy
Gravitational Potential Energy

Gravitational Potential Energy is the
energy associated with the relative
position of an object in space near the
Earth’s surface
 Objects interact with the earth through the
gravitational force
 Actually the potential energy of the earthobject system
Work and Gravitational Potential
Energy

Consider block of mass m at initial height yi

Work done by the gravitational force
W grav   F cos    y  ( mg cos  )  y , but :
 y  yi  y f , cos   1,
Thus : W grav  mg  yi  y f   mgyi  mgy f .
This quantity is called potential energy:
PE  m gy
 Note:
W
g ra vity
 PEi  PE
f
Important: work is related to the difference in PE’s!
Reference Levels for
Gravitational Potential Energy

A location where the gravitational potential
energy is zero must be chosen for each
problem
 The choice is arbitrary since the change in the
potential energy is the important quantity
 Choose a convenient location for the zero
reference height
○ often the Earth’s surface
○ may be some other point suggested by the problem
Reference Levels for
Gravitational Potential Energy
 A location
where the
gravitational potential energy is
zero must be chosen for each
problem
The choice is arbitrary since
the change in the potential
energy gives the work done

Wgrav1  mgyi1  mgy f1 ,
Wgrav2  mgyi2  mgy f 2 ,
Wgrav3  mgyi3  mgy f 3 .
Wgrav1  W grav2  Wgrav3 .
Potential Energy Stored in a
Spring
Involves the spring constant (or
force constant), k
 Hooke’s Law gives the force

F=-kx
○ F is the restoring force
○ F is in the opposite direction of x
○ k depends on how the spring was formed, the
material it is made from, thickness of the wire, etc.
Spring Example
Spring is slowly stretched
from 0 to xmax
 Fapplied = -Frestoring = kx
 W = {area under the curve} =
½(kx) x = ½kx²

Potential Energy
Example 6.31
 A 55-kg hiker starts at an elevation of
1600 m and climbs to the top of a 3300m peak. (a) What is the hiker’s change
in potential energy? (b) What is the
minimum work required of the hiker? (c)
Can the actual work done be more than
this? Explain why.

KE and Work-Energy Theorem
There are two general types
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
Conservative Forces

A force is conservative if the work it
does on an object moving between two
points is independent of the path the
objects take between the points
 The work depends only upon the initial and
final positions of the object
 Any conservative force can have a potential
energy function associated with it
Note: a force is conservative if the work it does on an object moving
through any closed path is zero.
Examples of Conservative Forces:

Examples of conservative forces
include:
 Gravity
 Spring force
 Electromagnetic forces

Since work is independent of the path:

W c  PEi  PE
f
: only initial and final points
KE and Work-Energy Theorem

Conservative Forces
 Work done is independent of path
1
Wf
Wi
2

Examples of conservative forces
include:
 Gravity
 Spring force
 Electromagnetic forces

Potential energy is another way of
looking at the work done by
conservative forces
Nonconservative Forces
A force is nonconservative if the work it
does on an object depends on the path
taken by the object between its final and
starting points.
 Examples of nonconservative forces

 kinetic friction, air drag, propulsive forces
KE and Work-Energy Theorem
Nonconservative Forces
 Work done is dependent of path
1
Wi
 Examples
include:
2
of nonconservative forces
 Kinetic friction
 Air drag
 Energy
 Heat
Wf
is transformed
Example: Friction as a
Nonconservative Force

The friction force transforms kinetic
energy of the object into a type of
energy associated with temperature
 the objects are warmer than they were
before the movement
 Internal Energy is the term used for the
energy associated with an object’s
temperature
Friction Depends on the Path
The blue path is
shorter than the
orange path
 The work required
is less on the blue
path than on the red
path
 Friction depends on
the path and so is a
nonconservative
force

KE and Work-Energy Theorem

To account for conservative and
nonconservative forces, the work-energy
theorem can be rewritten as
Wc + Wnc = ΔKE
Conservation of Mechanical
Energy

Conservation in general
 To say a physical quantity is conserved is to say
that the numerical value of the quantity remains
constant

In Conservation of Energy, the total
mechanical energy remains constant
 In any isolated system of objects that interact
only through conservative forces, the total
mechanical energy of the system remains
constant.
Conservation of Energy

Total mechanical energy is the sum of
the kinetic and potential energies in the
system
Ei  E f
K Ei  P Ei  K E f  P E f
 Other types of energy can be added to
modify this equation
Systems and Energy Conservation

Recall
Wnc + Wc = ΔKE

From Work-Energy Theorem
Wnc = ΔKE + ΔPE
= (KEf – KEi) + (PEf – PEi)
= (KEf + PEf) - (KEi + PEi)
= Ef – Ei = ΔE
Systems and Energy Conservation

Principle of Conservation of Energy
 In an isolated system, energy may be
transferred from one type to another, but the
total Etot of the system always remains
constant.
ΔEtot = ΔKE + ΔPE + ΔEint (changes in other
forms of energy) = 0
 If the system is not isolated then ≠ 0
 “Energy cannot be created or destroyed”
Systems and Energy Conservation

Conservation of Energy
 Consider a simple pendulum system
○ ΔEtot = ΔKE + ΔPE = 0
 At the highest points, all PE
○ Velocity = 0
○ Δy is maximum
 At the bottom, all KE
○ Velocity is maximum
○ Δy = 0
V =0
V =Vmax
Energy Conversion/Conservation Example
10 m
P.E. = 98 J
K.E. = 0 J

Drop 1 kg ball from 10 m
 starts out with mgh = (1 kg)(9.8 m/s2)(10
8m
P.E. = 73.5 J
K.E. = 24.5 J
6m
P.E. = 49 J
K.E. = 49 J
4m
2m
0m
m) = 98 J of gravitational potential energy
 halfway down (5 m from floor), has given up
half its potential energy (49 J) to kinetic
energy
○ ½mv2 = 49 J  v2 = 98 m2/s2  v  10 m/s
 at floor (0 m), all potential energy is given
P.E. = 24.5 J
K.E. = 73.5 J
P.E. = 0 J
K.E. = 98 J
up to kinetic energy
○ ½mv2 = 98 J  v2 = 196 m2/s2  v = 14 m/s
ConcepTest 6.16 Down the Hill
Three balls of equal mass start from rest and roll down different
ramps. All ramps have the same height. Which ball has the greater
speed at the bottom of its ramp?
4) same speed
for all balls
1
2
3
ConcepTest 6.16 Down the Hill
Three balls of equal mass start from rest and roll down different
ramps. All ramps have the same height. Which ball has the greater
speed at the bottom of its ramp?
4) same speed
for all balls
1
2
3
All of the balls have the same initial gravitational PE, since they
are all at the same height (PE = mgh). Thus, when they get to
the bottom, they all have the same final KE, and hence the same
speed (KE = 1/2 mv2).
Follow-up: Which ball takes longer to get down the ramp?
Systems and Energy Conservation

The changes occur in all forms of
energy
 ΔK + ΔP + Δ(all other forms of energy) = 0

Newton’s Laws fails in submicroscopic
world of atoms and nuclei
 Law of Conservation of Energy still applies!
○ One of the great unifying principles of science
Spring Potential Energy
Conservation of Energy with Spring
 The PE of the spring is added to both
sides of the conservation of energy
equation, Wnc = 0
(KE + PEg+PEs)i = (KE + PEg + PEs)f

The same problem-solving strategies
apply
Systems and Energy Conservation

Nonconservation of Energy
 Force is not conservative
○ Thermal energy from friction
 Internal Energy
 Positive work
○ Energy transferred from environment to
system
 Negative work
○ Energy transferred from system to
environment
Nonconservative Forces with
Energy Considerations
When nonconservative forces are
present, the total mechanical energy of
the system is not constant
 The work done by all nonconservative
forces acting on parts of a system
equals the change in the mechanical
energy of the system


W
n o n c o n s e r va t i v e
  E n erg y
Nonconservative Forces and
Energy

In equation form:
W n c  K E f  K E i  ( P E i  P E f ) or
W nc  ( K E f  P E f )  ( K Ei  P Ei )
The energy can either cross a boundary or
the energy is transformed into a form not
yet accounted for
 Friction is an example of a nonconservative
force

Transferring Energy

By Work
 By applying a force
 Produces a
displacement of the
system
Transferring Energy

Heat
 The process of
transferring heat by
collisions between
molecules
Transferring Energy

Mechanical Waves
 a disturbance
propagates through a
medium
 Examples include
sound, water, seismic
Transferring Energy

Electrical transmission
 transfer by means of
electrical current
Transferring Energy

Electromagnetic
radiation
 any form of
electromagnetic waves
○ Light, microwaves, radio
waves
Systems and Energy Conservation
 Energy Units
 Electron volts
- An electron volt is the amount of work done on
an electron when it moves through a potential
difference of one volt
○ Describes energies of atoms and molecules
○ 1 electron volt = 1.60217646 × 10-19 joules
 Examples
 Energy for the dissociation of an NaCl molecule into
Na+ and Cl- ions: 4.2 eV
 Ionization energy of atomic hydrogen: 13.6 eV
 Calories
- The energy needed to increase the temperature
of 1 g of water by 1 °C
○ Labels on food refer to kilocalories
○ 1 calorie = 4.18400 joules
Notes About Conservation of
Energy

We can neither create nor destroy energy
 Another way of saying energy is conserved
 If the total energy of the system does not remain
constant, the energy must have crossed the
boundary by some mechanism
 Applies to areas other than physics
Work-Energy Theorem

Example
Waterslides are nearly frictionless, and one such
slide named Der Stuka, is 72.0 feet high, found at
Six Flags in Dallas, Texas and Wet’n Wild in
Orlando, Florida. a) Determine the speed of a
60kg woman at the bottom of such a slide,
assuming no friction. b) If the woman is clocked at
18.0 m/s at the bottom of the slide, how much
mechanical energy was loss to friction?
Systems and Energy Conservation
 Example:
Problem
A bead of mass m =
5.00kg is released from
point A and slides on the
frictionless track shown
in the figure. Determine
a) the bead‘s speed at B
and C and b) the net
work done by the force
of gravity in moving the
bead from A to C.
Power
Often also interested in the rate at which
the energy transfer takes place
 Power is defined as this rate of energy
transfer


W
P 
 Fv
t
 SI units are Watts (W)
○
J kg  m 2
W  
s
s2
Power

US Customary units are generally hp
 Need a conversion factor
○ 1 hp = 550 (ft x lb)/s = 746 W
 Can define units of work or energy in terms
of units of power:
○ 1kWh = (103J/s)(3600s) = 3.60 x 106 J
 kilowatt hours (kWh) are often used in electric bills
 This is a unit of energy, not power
Power
Example:
 A 1.00x 103 kg elevator carries a
maximum load of 8.00 x 102kg. A
frictional force of 4.00 x 103 N impedes
the motion upward. What minimum
power (in kW and hp), must the motor
deliver to lift the full elevator at constant
speed of 3.00 m/s?
