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Chapter 07: Kinetic Energy and Work
Conservation of Energy is one of Nature’s fundamental
laws that is not violated.
Energy can take on different forms in a given system.
This chapter we will discuss work and kinetic energy.
Next chapter we will discuss potential energy.
If we put energy into the system by doing work, this
additional energy has to go somewhere. That is, the
kinetic energy increases or as in next chapter, the
potential energy increases.
The opposite is also true when we take energy out of a
system.
Different forms of energy
Kinetic Energy:
linear motion
rotational motion
Potential Energy:
gravitational
spring compression/tension
electrostatic/magnetostatic
chemical, nuclear, etc....
Mechanical Energy is the sum of Kinetic energy +
Potential energy. (reversible process)
Friction will convert mechanical energy to heat.
Basically, this (conversion of mechanical energy to heat
energy) is a non-reversible process.
Forms of energy
Kinetic energy: the energy associated with motion.
Potential energy: the energy associated with position
or state. (Chapter 08)
Heat: thermodynamic quanitity related to entropy.
(Chapter 20)
Conservation of Energy is one of Nature’s fundamental
laws that is not violated.
That means the grand total of all forms of energy in a
given system is (and was, and will be) a constant.
Chapter 07: Kinetic Energy and Work
Kinetic Energy is the energy associated with the motion
of an object.
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1
2
K = mv
m: mass and v: speed
Note: This form of K.E. holds only for speeds v << c.
SI unit of energy:
1 joule = 1 J = 1 kg.m2/s2
Other useful unit of energy is the electron volt (eV)
1 eV = 1.602 x 10-19J
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Work
Work is energy transferred to or from an object by
means of a force acting on the object.
Formal definition:
K K
W = ∫ F ⋅ ds
*Special* case: Work done by a constant force:
W = ( F cos φ) d = F d cos φ
What about the work done on an object moving with
constant velocity?
constant velocity => acceleration = 0 => force = 0
Consider 1-D motion.
 dv 
m  dx
xi
xi
i
 dt 
xf
xf
 dv 
 dv dx 
dx = ∫ mv dx
= ∫ m

xi
xi
 dx 
 dx dt 
vf
vf
1
1
1
= ∫ mvdv = mv2 | = mvf2 − mvi2 = K f − K i
vi
vi
2
2
2
xf
W = ∫ F dx = ∫
xf
(m a ) dx = ∫x
xf
So, kinetic energy is mathematically connected to
work!!
=> work = 0
Work-Kinetic Energy Theorem
Work done by a constant force:
W = F d cos φ = F . d
The change in the kinetic energy of a particle is equal the
net work done on the particle.
∆K = K f − K i = Wnet
1
1
= mv f2 − mvi2
2
2
.... or in other words,
Final kinetic energy = Initial kinetic energy + net Work
Kf =
1
1
mv f2 = K i + Wnet = mvi2 + Wnet
2
2
Consequences:
when φ < 90o , W is positive
when φ > 90o , W is negative
when φ = 90o , W = 0
when F or d is zero, W = 0
Work done on the object by the force:
–Positive work: object receives energy
–Negative work: object loses energy
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• W = F d cos φ = F . d
Question: What about circular motion?
• SI Unit for Work
1 N . m = 1 kg . m/s2 . m = 1 kg . m2/s2 = 1 J
• Net work done by several forces
Σ W = Fnet . d = ( F1 + F2 + F3 ) . d
= F1 . d + F2 . d + F3 . d = W1 + W2 + W3
v
a
Remember that the above equations hold ONLY for
constant forces. In general, you must integrate the
force(s) over displacement!
How much work was done and when?
A particle moves along the x-axis. Does the kinetic
energy of the particle increase, decrease , or remain the
same if the particle’s velocity changes?
Work done by Gravitation Force
(a) from –3 m/s to –2 m/s?
|vf | < |vi| means Kf < Ki,
so work is negative
(b) from –2 m/s to 2 m/s?
−2 m/s
0m/s: work negative
0 m/s
+2m/s: work positive
Together they add up to zero work.
Wg = mg d cos φ
throw a ball upwards:
– during the rise,
mg
Wg = mg d cos180o = −mgd
negative work
K & v decrease
– during the fall,
mg
Wg = mgd cos0o = mgd
positive work
K & v increase
d
d
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Sample problem 7-5: An
initially stationary 15.0 kg crate
is pulled a distance L = 5.70 m
up a frictionless ramp, to a height
H of 2.5 m, where it stops.
(a) How much work Wg is done on
the crate by the gravitational force
Fg during the lift?
Work done by a variable force
∆Wj = Fj, avg ∆x
W = Σ∆Wj = ΣFj, avg ∆x = lim ΣFj, avg ∆x
∆x →0
xf
W = ∫ F( x )dx
xi
Three dimensional analysis
xf
yf
zf
xi
yi
zi
W = ∫ Fx dx + ∫ Fy dy + ∫ Fz dz
Work done by a spring force
The spring force is given by
F = −kx ( Hooke’s law)
k: spring (or force) constant
k relates to stiffness of spring;
unit for k: N/m.
Spring force is a variable force.
x = 0 at the free end of
Work done by a spring force
The spring force tries to restore the system to its
equilibrium state (position).
Examples:
• springs
• molecules
• pendulum
Motion results in Simple Harmonic Motion (Chapter 15)
the relaxed spring.
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Work done by Spring Force
Work done by the spring force
xf
xf
xi
xi
Ws = ∫ Fdx = ∫ (−kx )dx =
1 2 1 2
kx i − kx f
2
2
If |xf| > |xi| (further away from x = 0); W < 0
If |xf| < |xi| (closer to x = 0); W > 0
If xi = 0, xf = x
then Ws = − ½ k x2
This work went into potential energy, since the
speeds are zero before and after.
The work done by an applied force (i.e., us):
If the block is stationary both before and after
the displacement:
vi = vf = 0
then Ki = Kf = 0
work-kinetic energy theorem: Wa + Ws = ∆K = 0
therefore:
Wa = − W s
Problem 07-74
A force directed along the
positive x direction acts on a
particle moving along the xaxis. The force is of the form:
F = a/x2, with a = −9.0 Nm2
Find the work done on the particle by the force as the
particle moves from x=1.0m to x = 3.0m.
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