Lecture 4
WORK and ENERGY
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OUTLINE
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Work and Kinetic Energy
The Work-Kinetic Energy Theorem
Power
Conservative Force-Nonconservative Force
Potential Energy
Mechanical Energy
Conservation of Mechanical Energy
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What is Energy?
• Energy is the power derived from the utilization of physical or
chemical resources, especially to provide light and heat or to
work machines.
Capability to produce the work.
• Technically, energy is a scalar quantity associated with the state (or
condition) of one or more objects
• Energy is a number that we associate with a system of one or more
objects. If a force changes one of the objects by, say, making it
move, then the energy number changes. ==> making machine etc...
• Energy can be transformed from one type to another and transferred
from one object to another, but the total amount is always the same
(energy is conserved). No exception to this principle of energy
conservation has ever been found.
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4.1 Work and Kinetic Energy
The work done by a constant force F on the object when it
moves a straight distance s is:
F=const

W = Fs cos = F.s

In general case, the work is not constant, the path is a curve.
s
The work done by force F when the object moves a very small
displacement ds (we can consider F constant and ds a straigh

linet:

F

ds
(1)
(2)
 
dv
dW = F .ds = m .ds = mv dv
dt
The work done by force F when the object moves from position (1) to (2) is:

v2

  1  1 
W =  F .ds =  mv dv = mv22 − mv12

2
2
1
v1
2
We define: Kinetic Energy:
1
K = mv2
2
The total work done on a particle is equal to the change in its kinetic energy
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1
W = K = mv22 − mv12
Work-Kinetic Energy Theorem
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2
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Kinetic energy
Kinetic energy K is energy associated with the state of motion of an object.
The faster the object moves, the greater is its kinetic energy.
When the object is stationary, its kinetic energy is zero.
For an object of mass m whose speed v is well below the speed of light
1 2
K = mv
2
The SI unit of kinetic energy (and every other type of energy) is the joule (J)
1 joule = 1 J = 1 kg.m2/s2.
Ex: a 3.0 kg duck flying pass us at 2.0 m/s has a kinetic energy of 6.0 kg m2/s2 = 6.0 J
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Work
Accelerate an object to greater speed by force increase the kinetic energy
Decelerate an object to slower speed by force decrease the kinetic energy
→The exerted force has transferred energy to the object from yourself or
from the object to yourself.
In such a transfer of energy via a force, work W is said to be done on
the object by the force
Work, W, is energy transferred to or from an object by means of a
force acting on the object.
Energy transferred to the object is positive work, and energy transferred
from the object is negative work.
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The Scalar Product (Dot product)
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4.2 Power
work _ done
Power =
per _ unit _ of _ time
t2
W =  dW =  Pdt
The power due to a force is the rate at
which that force does work on an object. If
the force does work W in an amount of
time t, the average power due to the
force over that time interval is
t1
if P = const = W = Pt
1 2 1 2
W = mv2 − mv1
2
2
1 2 1 2
mv2 − mv1
W 2
2
t = =
P
P
The instantaneous power P is the
instantaneous time rate of doing work

dW F .ds  
P=
=
= F .v
dt
dt
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Conservative
Nonconservative Force
1. Definition
• A force is Conservative if the work done by
the force is independent on the path, it is
dependent only on the initial and final
position.
• Gravity and spring force are conservative
forces, while kinetic friction is not.
2. Work done by Gravitation Force:
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(1)
r1
Fgrv
dr

m
ds
Work done by the grav. force F on object
m when it moves a displacement ds:

Mm
dW = F .ds = − F .ds
.
cos

=
−
G
dr



2
r
dr
(2)Work done by the grav. force F on object
m when it moves from (1) to (2)
r
r2
2
M
G = 6.67 10 −11 N .m 2 / kg 2
Mm
Mm
Mm
W =  − G 2 dr = G
−G
r
r2
r1
r1
Universal Gravitational constant
+ Work done is independent of the path, but of the initial and final position
 gravitation force is conservative.
+ we define a scalar quantity called gravitational potential energy of two object
separated by a distance r :
Mm
U (r ) = −G
r
+C
If we choose U= 0 when r=, we have C=0, If we choose U=0 on the
surface of Earth: C=GMm/R
We can write: W = U1 − U 2 = −U
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Spring Force
Where 𝑑Ԧ is the displacement of the spring’s free end from its position when the spring is in
its relaxed state (neither compressed nor extended), and k is the spring constant (a
measure of the spring’s stiffness). If an x axis lies along the spring, with the origin at the
location of the spring’s free end when the spring is in its relaxed state
A spring force is thus a variable force: It varies with the displacement of the spring’s free end.
Work Done by a Spring Force If an object is attached to the spring’s free end, the work Ws
done on the object by the spring force when the object is moved from an initial position xi to a
final position xf is
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Spring Force
The force exerted by a spring on a block varies with the block’s position x relative to
the equilibrium position x = 0.
When x is positive (stretched spring), the spring
force is directed to the left
When x is zero (natural length of the spring), the
spring force is zero.
When x is negative (compressed spring), the
spring force is directed to the right.
Ex: if the spring has a force constant of 80 N/m and is compressed 3.0 cm from
equilibrium, the work done by the spring force as the block moves from xi= - 3.0 cm
to its unstretched position xf=0 is 3.6 .10-2 J.
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Elastic potential Energy
Elastic potential energy is the energy associated with the state of compression
or extension of an elastic object.
For a spring that exerts a spring force F= -kx when its free end has displacement
x,the elastic potential energy is
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Conservative Forces
1. The work done by a conservative force on a particle moving
between any two points is independent of the path taken by
the particle.
2. The work done by a conservative force on a particle moving
through any closed path is zero. (A closed path is one in
which the beginning and end points are identical.)
3. the work Wc done by a conservative force on an object as
the object moves from one position to another is equal to the
initial value of the potential energy minus the final value.
(2)
(a)

W12 =  Fc. .ds =
(1a 2 )
F
(b)
(1)
ds
(1) (2)

 Fc..ds = U1 − U 2 = −U
(1b 2 )

W =  Fc. .ds = 0
(C )
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Mechanical Energy
  
ma = Fc + Fnc
• If an object is exerted by
Conservative Force Fc
and Nonconservative
Force Fnc,
• from the Work-Kinetic
Energy Theorem :
• Fc is conservative:
WFc = −U = U1 − U 2 (2)
• From (1) and (2):
K 2 − K1 = U1 − U 2 + WFnc
K = K 2 − K1 = WFc + WFnc (1)
( K 2 + U 2 ) − ( K1 + U1 ) = WFnc
• Mechanical Energy:
E=K+U
E = E2 − E1 = WFnc
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Conservation of Mechanical Energy
The change in Mechanical energy of an object
is equal to the work done by nonconservative
force on the object as it takes a path form
position (1) to (2)
E = E2 − E1 = WFnc
If Fnc= 0 or WFnc =0 => E=0: E=const
Conservation of Mechanical Energy
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Conservative Force & Potential Energy







F = Fx i + Fy j + Fz k ds = dxi + dyj + dzk

(1)
dW = Fc .ds = Fx dx + Fy dy + Fz dz
F _ is _ conservative
 U
U 
U
dz (2)
dy +
dx +
dW = −dU = −
z 
y
 x
U
U
U
; Fz = −
; Fy = −
Fx = −
z
y
x

F = − gradU
Gradient Operator
     
grad = i +
j+ k
x
y
z
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Potential Energy and
Equilibrium in One Dimension
For a general conservative force in one dimension,


F = Fx i
dU
Fx = −
dx
A particle is in equilibrium if the net
force acting on it is zero.
dU/dx=0=>U=min or Max
Stable Equilibrium: Umin
Unstable Equilibrium: Umax
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Relation between Consevativ e Force and Potential Energy

F = − gradU
U
U
U
Fx = −
; Fy = −
; Fz = −
x
y
z
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Problem 1
1. A particle moves along the x axis under the influence of a stationary object.
The net force on the particle is given by F = (8N/m 3 )x 3 .
If the potential energy is taken to be zero for x = 0 then the
potential energy is given by :
A. (2 J/m 4 )x 4 B. (-2J/m 4 )x 4
C. (24 J/m 2 x 2 ) D. (-24 J/m 2 )x 2
E. 5 J - (2 J/m 4 )x 4
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Problem 1
1. A particle moves along the x axis under the influence of a stationary object.
The net force on the particle is given by F = (8N/m 3 )x 3 .
If the potential energy is taken to be zero for x = 0 then the
potential energy is given by :
A. (2 J/m 4 )x 4 B. (-2J/m 4 )x 4
C. (24 J/m 2 x 2 ) D. (-24 J/m 2 )x 2
E. 5 J - (2 J/m 4 )x 4
dU
dx
= −dU = Fdx
F =−
U ( x)
−
x
 dU =  8x
U ( x =0 ) =0
−U = 2x4
3
dx
ans : B
0
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Problem 2
A 0.20 - kg particle moves along the x axis
under the influence of a stationary object.
The potential energy is given by
U(x) = (8.0J/m2 )x 2 + (2.0J/m4 )x 4 ,
where x is in coordinate of the particle.
If the particle has a speed of 5.0m/s
when it is at x = 1.0m,
its speed when it is at the origin is :
A. 0 B. 2.5m/s
C. 5.7m/s
D. 7.9m/s
E. 11m/s
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Problem 2
A 0.20 - kg particle moves along the x axis
under the influence of a stationary object.
The potential energy is given by
Given U = 8x 2 + 2x 4
U(x) = (8.0J/m )x + (2.0J/m )x ,
where x is in coordinate of the particle.
If the particle has a speed of 5.0m/s
KE at x = 1m : T =
2
2
4
4
when it is at x = 1.0m,
its speed when it is at the origin is :
A. 0 B. 2.5m/s
C. 5.7m/s
D. 7.9m/s
E. 11m/s
ans : E
PE at x = 1m : U = 8  12 + 2  14 = 10J
1
1
mv 2 =  0.2  52 = 2.5J
2
2
Mechanical Energy at x = 1m : E = U + T = 12.5J
The force exerting on the object
is conservati ve force,
so there is Conservata tion of Mech. Energy.
E x =0 = E x =1m = E
1
E x =0 = U x =0 + T x =0 = 0 + mv o2
2
2E
2  12.5
vo =
=
= 11.2m/s
m
0.2
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