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Chapter 7
Lecture 13:
Kinetic Energy and Work: II
HW5 (problems): 7.4, 7.15, 7.17,
7.31, 7.46, 7.63, 8.9, 8.31
Due Friday, March 10.
Work Done by a Varying Force



Assume that during a very
small displacement, Dx, F
is constant
For that displacement,
W ~ F Dx
For all of the intervals,
xf
W   Fx Dx
xi
Work Done by a Varying Force,
cont

lim
Dx 0
xf
 F Dx  
x
xi

xf
xi
Fx dx
xf
Therefore,W 
 Fx dx
xi

The work done is equal
to the area under the
curve between xi and xf
Work Done By Multiple Forces

If more than one force acts on a system and
the system can be modeled as a particle, the
total work done on the system is the work
done by the net force
W  W
net


xf
xi
  F dx
x
In the general case of a net force whose
magnitude and direction may vary
W  W
net

xf
xi
  Fdr
Work Done by Multiple Forces,
cont.

If the system cannot be modeled as a
particle, then the total work is equal to the
algebraic sum of the work done by the
individual forces
Wnet  Wby individual forces

Remember work is a scalar, so this is the
algebraic sum
Work Done By A Spring



A model of a common
physical system for
which the force varies
with position
The block is on a
horizontal, frictionless
surface
Observe the motion of
the block with various
values of the spring
constant
Hooke’s Law

The force exerted by the spring is
Fs = - kx



x is the position of the block with respect to the equilibrium
position (x = 0)
k is called the spring constant or force constant and measures
the stiffness of the spring
This is called Hooke’s Law
Hooke’s Law, cont.



When x is positive
(spring is stretched), F
is negative
When x is 0 (at the
equilibrium position), F
is 0
When x is negative
(spring is compressed),
F is positive
Hooke’s Law, final



The force exerted by the spring is always
directed opposite to the displacement from
equilibrium
The spring force is sometimes called the
restoring force
If the block is released it will oscillate back
and forth between –xmax and xmax
Work Done by a Spring


Identify the block as the system
Calculate the work as the block moves from xi = - xmax to xf = 0
xf
0
xi
 xmax
Ws   Fx dx  

 kx  dx 
1 2
kxmax
2
The total work done as the block moves from
–xmax to xmax is zero
Work Done by a Spring, cont.


Assume the block undergoes an arbitrary
displacement from x = xi to x = xf
The work done by the spring on the block is
Ws  
xf
xi

1 2 1 2
 kx  dx  kxi  kxf
2
2
If the motion ends where it begins, W = 0
Kinetic Energy

Kinetic Energy is the energy of a particle due
to its motion

K = ½ mv2




K is the kinetic energy
m is the mass of the particle
v is the speed of the particle
A change in kinetic energy is one possible
result of doing work to transfer energy into a
system
Kinetic Energy, cont

Calculating the work:
W 
xf
xi
 F dx  
xf
xi
ma dx
vf
W   mv dv
vi
1 2 1
2
W

mv

mv

f
i
2
2
Wnet  K f  K i  DK
7.8: Work kinetic energy theorem with a variable force
A particle of mass m is moving along an x axis and acted on by a net force
F(x) that is directed along that axis.
The work done on the particle by this force as the particle moves from
position xi to position xf is :
But,
Therefore,
Work-Kinetic Energy Theorem


The Work-Kinetic Energy Theorem states SW = Kf –
Ki = DK
When work is done on a system and the only
change in the system is in its speed, the work done
by the net force equals the change in kinetic energy
of the system.



The speed of the system increases if the work done on it is
positive
The speed of the system decreases if the net work is
negative
Also valid for changes in rotational speed
Work-Kinetic Energy Theorem
– Example



The normal and
gravitational forces do no
work since they are
perpendicular to the
direction of the
displacement
W = F Dx
W = DK = ½ mvf2 - 0
Sample problem, industrial spies
7.6: Work done by gravitational force
(a) An applied force lifts an object.
The object’s displacement makes an
angle f =180° with the gravitational
force on the object. The applied force
does positive work on the object.
(b) An applied force lowers an object. The
displacement of the object makes an angle
with the gravitational force .The applied
force does negative work on the object.
7.8: Work done by a general variable force
B. Three dimensional force:
If
where Fx is the x-components of F and so on,
and
where dx is the x-component of the displacement vector dr and so on,
then
Finally,
Instantaneous Power


Power is the time rate of energy transfer
The instantaneous power is defined as
dE

dt

Using work as the energy transfer method,
this can also be written as
avg
W

Dt
Power


The time rate of energy transfer is called
power
The average power is given by
W
P
Dt

when the method of energy transfer is work
Instantaneous Power and
Average Power

The instantaneous power is the limiting value
of the average power as Dt approaches zero
dW
dr
lim W
 Dt 0

 F
 F v
Dt
dt
dt

The power is valid for any means of energy
transfer
Instantaneous Power and
Average Power

The SI unit of power is called the watt



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1 watt = 1 joule / second = 1 kg . m2 / s3
A unit of power in the US Customary system is
horsepower
1 hp = 746 W
Units of power can also be used to express
units of work or energy

1 kWh = (1000 W)(3600 s) = 3.6 x106 J
Potential Energy

Potential energy is energy related to the
configuration of a system in which the
components of the system interact by forces


The forces are internal to the system
Can be associated with only specific types of
forces acting between members of a system
8.2 Work and potential energy
The change DU in potential energy
(gravitational, elastic, etc) is defined as
being equal to the negative of the work
done on the object by the conservative
force (gravitational, elastic, etc)
Gravitational Potential Energy


The system is the Earth
and the book
Work is done on the
book by lifting it slowly
a vertical displacement
D r  Dyˆj

The work done on the
system must appear as
an increase in the
energy of the system
Gravitational Potential Energy,
cont


There is no change in kinetic energy since
the book starts and ends at rest
Gravitational potential energy is the energy
associated with an object at a given location
above the surface of the Earth
Gravitational Potential Energy,
final

The quantity mgy is identified as the
gravitational potential energy, Ug



Ug = mgy
Units are joules (J)
Is a scalar