Download lecture 14 conservation of energy

LECTURE 14
CONSERVATION OF ENERGY
Instructor: Kazumi Tolich
Lecture 14
2
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Reading chapter 8-4 to 8-5
¤ Conservation
of energy
¤ Potential energy curves and equipotentials
Quiz: 1
3
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A student wants to lift a 50-kg mass, so she
hooks it up to the pulley system as shown. The
resulting force required to lift the mass is now
half the weight of the mass. If the mass rises
by 10 cm, how much work did the student do
pulling the rope at a constant speed in Joules?
Mass
Quiz: 14-1 answer
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50 J
If the mass is raised by 10 cm, then both of the right ropes are now
10 cm shorter. The total rope length must be the same, so the left rope is
20 cm longer. So the student pulled it by 20 cm.
Work = 𝐹𝑑 =
(
)
𝑑=
*+
)
𝑑=
,- /0 1.34 5⁄67
)
0.20 m = 50 J
This is equal to the change in mechanical energy of the Earth-mass
system.
∆𝐸 = ∆𝐾 + ∆𝑈 = 𝑚𝑔∆𝑦 = 50 kg 9.81 m⁄s ) 0.10 m = 50 J
Mass
Quiz: 2
5
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Suppose a book is sliding on a horizontal tabletop, slowing
down due to the friction between the book and the table
surface. Eventually the book stops. Define a system containing
the book and the table. Which of the following statements
is/are correct during this process? Choose all that apply.
A.
B.
The mechanical energy of the system is conserved.
The total energy of the system is conserved.
Quiz: 14-2 answer
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The total energy of the system is conserved.
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Initially, the book is moving, so the system has a non-zero kinetic energy. 𝐾 = 𝑚𝑣 )
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)
Finally, the book stops on top of the stationary table, so the system does not have kinetic
energy.
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The kinetic energy decreased to zero, so the mechanical energy is not conserved:
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∆𝐸 = ∆𝐾 + ∆𝑈 ≠ 0
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The reduced mechanical energy went to heat up the table and book surfaces through the
friction.
The energy associated with the temperature of a system is its internal energy.
Change in mechanical energy
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If in a system an object slides through a distance 𝑑 with a kinetic friction 𝑓/
applied on it, the mechanical energy of the system changes.
∆𝐸 = −𝑓/ 𝑑
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Other factors that changes mechanical energy:
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Air or water resistance
Energy to deform an object result in increase of thermal energy.
If chemical reaction takes place within the system, change in chemical energy occurs.
Other processes can change nuclear, electromagnetic, or other forms of energy.
Quiz: 3
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A body falls through the atmosphere gaining 20J of kinetic energy. Air
resistance is not negligible. If you define a system containing the body, its
surrounding air, and the earth, how much gravitational potential energy did
the system lose?
A.
B.
C.
20 J
more than 20 J
less than 20 J
Quiz: 14-3 answer
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more than 20 J
As the body falls, the gravitational potential energy is converted to kinetic energy
of the body as well as internal energy of the air and the body as they heat up.
−∆𝑈 = ∆𝐾 + ∆𝐸OPQ = 20 J + ∆𝐸OPQ
Example: 1
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At a playground, a child of mass
𝑚 = 18 kg plays on a slide that
drops through a height of ℎ = 2.2 m.
The child starts at rest at the top of
the slide. On the way down, the
friction causes the slide and child to
heat up, creating ∆𝐸OPQ = 373 J of
internal energy. What is the child’s
speed at the bottom of the slide?
Law of conservation of energy/Demo: 1
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The law of conservation of energy:
Within the isolated system, energy can be converted from one form to another, or
transmitted from one region to another, but energy can never be created or
destroyed.
The change in the total energy of a system is equal to the energy that enters minus
the energy that leaves the system.
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The law of conservation of energy is one of the most fundamental laws of physics.
We have never observed any violation of this law.
Demo: bowling ball pendulum
“Conserving” energy
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If energy cannot be created or destroyed, why do we bother
“conserving” energy by turning off unused lights, etc?
¤ There
are useful forms of energy and not so useful form (thermal energy).
Converting thermal energy to more useful form of energy does not have
100% efficiency.
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On “Earth Day,” many people tried to “conserve” electric energy by
turning off the lights and lighting candles. What is wrong with this
picture?
Example: 2
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A moving block with a mass 𝑚 = 2.5 kg
collides with a horizontal spring whose
spring constant is 𝑘 = 320 N/m. The block
compresses the spring a maximum distance
of 7.5 cm from its rest position. The
coefficient of kinetic friction between the
block and the horizontal surface is
𝜇/ = 0.25.
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a)
b)
c)
How much work is done by the spring in
bringing the block to rest?
How much mechanical energy is dissipated by
the force of friction while the block is being
brought to rest by the spring?
What is the speed of the block when it hits
the spring?
Gravitational potential energy curves
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A ball on a frictionless track starts at rest at A.
The ball-Earth system has the least potential energy, and the most
kinetic energy at B.
The ball turns around at D (turning point) and repeats the process.
2D gravitational potential energy plot
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Lines corresponding to constant values of
potential energy are called
equipotentials.
Since 𝑈 = 𝑚𝑔𝑦, where 𝑦 is the height
above where you define 𝑈 to be zero,
equipotential lines for gravitational
potential energy is at the same height.
Elastic potential energy curves
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A mass attached to a spring is pulled by distance 𝐴 from the
equilibrium position and released.
The mass undergoes an oscillatory motion.
1
𝑈 = 𝑘𝑥 )
2
Quiz: 4
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The graph shows the gravitational potential energy of an object-Earth system as a function of
position. When the object is moving in the negative 𝑥-direction at 𝑥 = 0.15 m, the kinetic
energy is 𝐾 = 30 J. What value of 𝑥 does the object change its direction of motion? Indicate
the position on the 𝒙-axis.
Quiz: 13-4 answer
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The total mechanical energy
is ~650 J.
𝐾 = 30 J
𝐸
𝑈
Turning point