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Chapter 2
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Chapter 3
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Chapter 4
71.6%
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6.7
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Chapter 5
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Chapter 6
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Homework & 2nd Midterm Exam
Proficiency: estimated by taking into account a number of
factors, such as answers to questions, confidence, and time.
Performance: the number of correct answers out of the number
of attempted questions.
Objects
Chapter 7
Kinetic energy and work
Sample problem: Vertical circular loop
Is the game safe?
vmin = gR
-  Why the water does not fall when
the bucket is upside down?
-  When the water starts to spill?
-  Does he need to swirl the bucket
faster than vmin when the bucket
has more water?
Do we need a higher
speed at the bottom
for bigger mass?
7.1 Some Physics Background
Dynamics: It is about the evolution of a system or process in time.
A conservation law in physics: A particular measurable property of an
isolated physical system does not change as the system evolves in time
Quantities that are conserved
-  Momentum
-  Angular momentum
-  Energy
-  Electric charge
-  Lepton number
-  …
Other conservation/invariant principles in dynamics
-  Parity
-  Time
-  …
7.2 What is energy?
One definition:
Energy is a scalar quantity associated with the state
(or condition) of one or more objects.
Major characteristics:
1.  Energy can be transformed from one form to another and transferred
from one object to another
2.  The total amount of energy is always the same (energy is conserved)
7.2 What is energy? Type and form
Forms of energy: We often hear
• 
• 
• 
• 
• 
Thermal energy
Chemical energy
Electric energy
Radiant energy
(electromagnetic
radiation)
•  Nuclear energy
•  Wind energy
•  Wave energy
Magnetic energy
•  Elastic energy
•  Sound energy
•  Mechanical
energy
•  Luminous energy
•  Mass (E=mc²)
Two types of most fundamental ones:
(1)  kinetic energy: a function of the movement of an object.
(2)  potential energy: a function of the position of an object in a field.
Looking at the microscopic, more fundamental level …
7.2 What is energy? Type and form
For example:
Thermal energy: It is the sum of the potential and kinetic energy on the
atomic and subatomic scale.
Chemical energy: In a macroscopic notion it is the sum of the potential
and kinetic energy on the atomic and subatomic scale.
Sound energy: Associated with the vibration or disturbance of matter.
…
7.2 What is energy? Energy transformation
Energy transformation all time around us, in our life:
Fire (Chemical energy → Heat and Light)
Friction (Kinetic energy → Heat)
Fuel cells (Chemical energy → Electric energy)
Heat engines (engine used in cars, Heat → Mechanical energy)
Heater (Electric energy → Heat)
Hydroelectric dams (Gravitational potential energy → Electric energy)
Electric generator (Kinetic energy or Mechanical work → Electric energy)
Battery (electricity) (Chemical energy → Electric energy)
Electric lamp (Electric energy → Heat and Light)
Microphone (Sound → Electric energy)
Windmills (Wind energy → Electric energy or Mechanical energy)
Piezoelectrics (Strain → Electric energy)
Ocean thermal power (Heat → Electric energy)
Geothermal power (Heat→ Electric energy)
Some are reversible transformations and others are non-reversible!
7.2 Conservation of Energy
The total amount of energy is always the same (energy is conserved)
- Energy can neither be created (produced) nor destroyed by itself. It
can only be transformed.
-  One of the most fundamental law of nature which we have never find
exceptions so far
-  Deep understanding: mathematical consequence of translational
symmetry of time.
-  Some historic cases which people once thought the energy is not
conserved.
-  Neutrinos
-  Dark energy
7.2 Applications of the law of conservation of energy
It is a very useful law!
-  It is a constraint that applies to all systems.
-  It can be used to solve problems : This will be the focus of this
chapter.
-  But we have to know more about energy before we can use this tool!
7.3 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.
For an object of mass m whose speed v is well below the
speed of light,
The SI unit of kinetic energy (and every other type of
energy) is the joule (J),
kg m/s
1 Joule = 1 J = 1 kgm2/s2.
Q: The graph on the right shows the velocity
of an object as a function of time. Which
of the graph of kinetic energy versus
time matches the velocity data?
A: The graph on the right shows the velocity
of an object as a function of time. Which
of the graph of kinetic energy versus
time matches the velocity data?
Sample Problem
What do we know?
•  a
•  x1-x0
•  Weight=mg
Sample Problem
7.4: Work
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.
Energy is needed to “do” work!
7.5: Work and kinetic energy
The work by a force F on an object through some displacement d:
-  The force component along the object s displacement.
-  The force component perpendicular to the displacement direction does zero
work.
For a constant force F, the work done W is:
7.5: Work and kinetic energy
When two or more forces act on an object, the net work done on the
object is the sum of the works done by the individual forces.
Since d is the same for all forces
!" !" !"
!"
W = (F 1 + F 2 + F 3 +...)• d
!" !" !" !" !" !"
= F 1 • d + F 2 • d + F 3 • d +...
= W1 + W2 + W3 +...
7.5: Work and kinetic energy
Two restrictions:
(1)  The force must be a constant force; that is, it must not change in
magnitude or direction as the object moves.
(2)  The object must be particle-like. This means that the object must be
rigid; all parts of it must move together, in the same direction.
Net Work:
When two or more forces act on an object, the net work done on the
object is the sum of the works done by the individual forces. Two ways
to calculate the net work:
(1)  Find the work done by each force and then sum those works.
(2)  First find the net force of those forces.
Q: A 5.0-kg ball on the end of a chain is whirled at a constant speed of
1.0 m/s in a horizontal circle of radius 3.0 m. What is the work
done by the centripetal force during one revolution?
a) 2.5 J
b) 1.7 J
c) 1.2 J
d) 0.56 J
e) zero J
A: A 5.0-kg ball on the end of a chain is whirled at a constant speed of
1.0 m/s in a horizontal circle of radius 3.0 m. What is the work
done by the centripetal force during one revolution?
a) 2.5 J
b) 1.7 J
c) 1.2 J
d) 0.56 J
e) zero J
A: A cart pushed by a man with constant force F. It remains constant
velocity v. Which statement(s) are right?
a) The man did no work.
b) The man did positive work to the cart
c) The man did negative work to the cart
d) The cart gained kinetic energy
e) In this process this is energy transformation among different forms
of energies.
A: A cart pushed by a man with constant force F. It remains constant
velocity v. Which statement(s) are right?
a) The man did no work.
b) The man did positive work to the cart
c) The man did negative work to the cart
d) The cart gained kinetic energy
e) In this process this is energy transformation among different forms
of energies.
March 18
1.  Michael Dowding: Canceled the Thursday, 3pm recitation
and referred those students to another recitation for this week.
2.  The answers for PHYS 211 Exam 2 are posted at my website.
3.  No class on April 10th (after the next in-class exam April 8)
due to 2015 SD AoS Annual Meeting.
7.5: Work and kinetic energy
Work-kinetic energy theorem
The theorem says that the change in kinetic energy of
a “particle” is the net work done on the particle.
Ekf - Eki = W
Attention
①  “Particle”
②  Other forms of energy: Thermal, chemical, etc.
③  Internal energy
7.5: Work and kinetic energy
Q: A particle moves along an x axis. Does the
kinetic energy of the particle increase, decrease, or
remain the same if the particle's velocity changes
1)  from -3 m/s to -2 m/s ?
2)  from -2 m/s to +2 m/s ?
3)  In each situation, is the work done on the particle
positive, negative, or zero?
7.5: Work and kinetic energy
Q: A particle moves along an x axis. Does the
kinetic energy of the particle increase, decrease, or
remain the same if the particle's velocity changes
1)  from -3 m/s to -2 m/s ?
2)  from -2 m/s to +2 m/s ?
3)  In each situation, is the work done on the particle
positive, negative, or zero?
(1)  decrease;
(2)  same;
(3)  (1) -- negative (2) -- zero
7.3.3. Ignoring friction effects, the amount of energy required to
accelerate a car from rest to a speed v is E. The energy is delivered
to the car by burning gasoline. What additional amount of energy
is required to accelerate the car to a speed 2v?
a) 0.5E
b) E
c) 2E
d) 3E
e) 4E
7.3.3. Ignoring friction effects, the amount of energy required to
accelerate a car from rest to a speed v is E. The energy is delivered
to the car by burning gasoline. What additional amount of energy
is required to accelerate the car to a speed 2v?
a) 0.5E
b) E
c) 2E
d) 3E
e) 4E
Sample problem, industrial spies
industrial spies …
industrial spies
Industrial spies
industrial spies …
By dynamics
Net force:
F = F1×cos30°+F2×cos40°=12.0×cos30°+10.0×cos40°
= 18.05 N
Acceleration:
a=F/m=18.05 N/225 kg = 0.08 m/s2
(2-16): v2 = v02 + 2a(x-x0)
V=[2×0.08×(8.50-0.0)]1/2 = 1.166 m/s !
Another sample problem: constant force in unit vector
notation
Another sample problem: constant force in unit vector
notation
Another sample problem: constant force in unit vector
notation
7.6: Work done by gravitational force: Two cases
7.6: Work done by gravitational force: Two cases
Eq. 7-12:
(a) An applied force lifts an
object. The displacement of the
object makes an angle φ =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.
Sample problem: accelerating
elevator cab
Sample problem: accelerating
elevator cab
Eq. 7-12: Wg=mgdcosΦ
Sample problem:
accelerating elevator cab
Sample problem:
accelerating elevator cab
Sample problem:
accelerating elevator cab
Eq. 7-1: Ek = mv2/2
Eq. 7-11: Kf =Ki +W
March 20
7.7: Work done by a spring force
Not a constant force!
Hooke s Law: To a good approximation for many springs, the force
from a spring is proportional to the displacement of the free end from its
position when the spring is in the relaxed state. The spring force is given
by Fs = − kx
(1) The “–” sign: the direction of the spring force is always opposite the
direction of the displacement of the spring s free end.
(2) The constant k: the spring constant (or force constant). It is a
measure of the stiffness of the spring.
7.7: Work done by a spring force
Fs = − kx
!
! 1!d
dWs = Fs • dD " "# = Fs dx = !kx $ dx
The net work Ws done by a spring, when it has a distortion from xi to
xf , is:
Work Ws :
(1) Positive if the block ends up closer to the relaxed position (x =0)
than it was initially.
(2) Negative if the block ends up farther away from x =0.
(3) It is zero if the block ends up at the same distance from x= 0.
Sample problem: work done by spring
Sample problem:
work done by spring
7.8: Work done by a general variable force
A. One-dimensional force, graphical analysis:
•  Divide the area under the curve of F(x) into a
number of narrow strips of width x.
•  Choose x small enough to permit us to take
the force F(x) as being reasonably constant over
that interval.
•  Let Fj,avg be the average value of F(x) within
the jth interval.
•  The work done by the force in the jth interval
is approximately
ΔW j = Fj ,avg Δx
⇒ W = ∑ ΔW j = ∑ Fj ,avg Δx
•  Wj is then equal to the area of the jth
rectangular, shaded strip.
7.8: Work done by a general variable force
A. One-dimensional force, calculus analysis:
Reducing the strip width Δx and
using more strips.  The
approximation better
The strip width approaches zero,
the number of strips then becomes
infinitely large  an exact result:
xf
ΔW = lim ∑ Fj ,avg Δx = ∫x F( x )dx
Δx→0
i
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,
7.8: Work kinetic energy theorem with a variable force
C: the factor of 1/2
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 :
since
Therefore,
Hello,
Ek needs you, “1/2”
Sample problem: work calculated from graphical method:
Sample problem: work calculated from graphical method:
Q: The block's kinetic energy at x1 is K1 = 280 J. What is the block's
speed at (1) x1=0, (2) x2=4.0 m, and (3) x3=6.5 m?
Sample problem: work calculated from graphical method:
(1) x1=0
Sample problem – cnt.
(2) x2=4.0 m
Sample problem – cnt.
(3) x3=6.5 m
March 23
Sample problem: work from 2-D integration:
1.  Variable force (both amplitude and direction)
2.  Two-dimensional motion
F
Sample problem: work from 2-D integration:
F
+19
-12
7.9: Power
The power due to a force: The time rate at which work is done by a
force.
If a force does an amount of work W in an amount of time t, the average
power due to the force during that time interval is
The instantaneous power P is the instantaneous time rate of
doing work, which we can write as
The SI unit of power is Joule/second, or Watt (W).
In the British system, the unit of power is the footpound
per second. Often the horsepower is used.
7.9: Power: in the case with constant force
1-D case:
(7-47)
3-D case:
(7-48)
7.9: Power: in the case with constant force
When two or more forces act on an object, the net work done on the
object is the sum of the works done by the individual forces.
!" !" !"
!"
W = (F 1 + F 2 + F 3 +...)• d
!" !" !" !" !" !"
= F 1 • d + F 2 • d + F 3 • d +...
= W1 + W2 + W3 +...
dW dW1 dW2 dW3
P=
=
+
+
...
dt
dt
dt
dt
= P1 + P2 + P3 +...
Sample problem: power, force, velocity:
Sample problem: power, force, velocity:
Sample problem: power, force, velocity:
1.  Positive power  the force is transferring energy to the
box at the rate of 6.0 J/s.
2.  Negative power  the force is taking energy from the box
at the rate of 6.0 J/s.
3.  The net power is the sum of the individual powers:
Pnet = P1 + P2= - 6.0 W +6.0 W= 0 !
(a)  The kinetic energy of the box is not changing
(b)  The speed of the box will remain at 3.0 m/s.  The net
power remains the same.
Some additional in-class
exercises
7.6.4. Consider the box in the drawing. We can slide the box up the frictionless incline from
point A and to point C or we can slide it along the frictionless horizontal surface from
point A to point B and then lift it to point C. How does the work done on the box along
path A-C,WAC, compare to the work done on the box along the two step path A-B-C,
WABC?
a) WABC is much greater than WAC.
b) WABC is slightly greater than WAC.
c) WABC is much less than WAC.
d) WABC is slight less than WAC.
e) The work done in both cases is the same.
7.6.4. Consider the box in the drawing. We can slide the box up the frictionless incline from
point A and to point C or we can slide it along the frictionless horizontal surface from
point A to point B and then lift it to point C. How does the work done on the box along
path A-C,WAC, compare to the work done on the box along the two step path A-B-C,
WABC?
a) WABC is much greater than WAC.
b) WABC is slightly greater than WAC.
c) WABC is much less than WAC.
d) WABC is slight less than WAC.
e) The work done in both cases is the same.
7.7.1. Block A has a mass m and block B has a mass 2m. Block A is pressed
against a spring to compress the spring by a distance x. It is then
released such that the block eventually separates from the spring and it
slides across a surface where the friction coefficient is µk. The same
process is applied to block B. Which one of the following statements
concerning the distance that each block slides before stopping is correct?
a) Block A slides one-fourth the distance that block B slides.
b) Block A slides one-half the distance that block B slides.
c) Block A slides the same distance that block B slides.
d) Block A slides twice the distance that block B slides.
e) Block A slides four times the distance that block B slides.
7.7.1. Block A has a mass m and block B has a mass 2m. Block A is pressed
against a spring to compress the spring by a distance x. It is then
released such that the block eventually separates from the spring and it
slides across a surface where the friction coefficient is µk. The same
process is applied to block B. Which one of the following statements
concerning the distance that each block slides before stopping is correct?
a) Block A slides one-fourth the distance that block B slides.
b) Block A slides one-half the distance that block B slides.
c) Block A slides the same distance that block B slides.
d) Block A slides twice the distance that block B slides.
e) Block A slides four times the distance that block B slides.
7.7.2. In designing a spring loaded cannon, determine the spring
constant required to launch a 2.0 kg ball with an initial speed of
1.2 m/s from a position where the spring is displaced 0.15 m from
its equilibrium position.
a) 16 N/m
b) 32 N/m
c) 64 N/m
d) 130 N/m
e) 180 N/m
7.7.2. In designing a spring loaded cannon, determine the spring
constant required to launch a 2.0 kg ball with an initial speed of
1.2 m/s from a position where the spring is displaced 0.15 m from
its equilibrium position.
a) 16 N/m
b) 32 N/m
c) 64 N/m
d) 130 N/m
e) 180 N/m
7.8.1. A 12 500-kg truck is accelerated from rest by a net force that
decreases linearly with distance traveled. The graph shows this
force. Using the information provided and work-energy methods,
determine the approximate speed of the truck when the force is
removed.
a) 8.41 m/s
b) 12.5 m/s
c) 17.7 m/s
d) 25.0 m/s
e) 35.4 m/s
7.8.1. A 12 500-kg truck is accelerated from rest by a net force that
decreases linearly with distance traveled. The graph shows this
force. Using the information provided and work-energy methods,
determine the approximate speed of the truck when the force is
removed.
a) 8.41 m/s
b) 12.5 m/s
c) 17.7 m/s
d) 25.0 m/s
e) 35.4 m/s
r
7.8.2. A net force given by F = 12 N + (1.4 x N) î is applied to an object
that is initially at rest. What is the change in the object s kinetic
energy as it moves from x1 = 0.50 m to x2 = 2.50 m?
a) 14 J
b) 17 J
c) 21 J
d) 24 J
e) 28 J
r
7.8.2. A net force given by F = 12 N + (1.4 x N) î is applied to an object
that is initially at rest. What is the change in the object s kinetic
energy as it moves from x1 = 0.50 m to x2 = 2.50 m?
a) 14 J
b) 17 J
c) 21 J
d) 24 J
e) 28 J
7.9.1. An SUV is accelerated from rest to a speed v in a time interval t.
Neglecting air resistance effects and assuming the engine is
operating at its maximum power rating when accelerating,
determine the time interval for the SUV to accelerate from rest to a
speed 2v.
a) 2t
b) 4t
c) 2.5t
d) 3t
e) 3.5t
7.9.1. An SUV is accelerated from rest to a speed v in a time interval t.
Neglecting air resistance effects and assuming the engine is
operating at its maximum power rating when accelerating,
determine the time interval for the SUV to accelerate from rest to a
speed 2v.
a) 2t
b) 4t
c) 2.5t
d) 3t
e) 3.5t
7.9.2. A television is rated at 450 W. What is the cost of operating the
TV for 5 hours, if the utility charges $0.085 per kilowatt-hour?
a) $0.12
b) $0.19
c) $0.43
d) $0.85
e) $1.91
7.9.2. A television is rated at 450 W. What is the cost of operating the
TV for 5 hours, if the utility charges $0.085 per kilowatt-hour?
a) $0.12
b) $0.19
c) $0.43
d) $0.85
e) $1.91
7.9.3. While you sleep, your body is using energy at a rate of 77 W.
How many food calories are used during an eight hour period?
One food calorie (C) is equal to 4186 joules.
a) 66 C
b) 240 C
c) 530 C
d) 710 C
e) 1200 C
7.9.3. While you sleep, your body is using energy at a rate of 77 W.
How many food calories are used during an eight hour period?
One food calorie (C) is equal to 4186 joules.
a) 66 C
b) 240 C
c) 530 C
d) 710 C
e) 1200 C