Transcript Work & Energy

Conservation of Energy
Chapter 11
Conservation of Energy
•
The Law of Conservation of Energy
simply states that:
1. The energy of a system is constant.
2. Energy cannot be created nor destroyed.
3. Energy can only change form (e.g. electrical to
mechanical to potential, etc).
– True for any system with no external forces.
ET = KE + PE + Q
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–
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KE = Kinetic Energy
PE = Potential Energy
Q = Internal Energy [kinetic energy due to the
motion of molecules (translational, rotational,
vibrational)]
Conservation of Energy
Energy
Mechanical
Kinetic
Nonmechanical
Potential
Gravitational
Elastic
Conservation of Mechanical Energy
• Mechanical Energy:
– If Internal Energy is ignored:
ME = KE + PE
• PE could be a combination of gravitational and
elastic potential energy, or any other form of
potential energy.
– The equation implies that the mechanical
energy of a system is always constant.
• If the Potential Energy is at a maximum, then
the system will have no Kinetic Energy.
• If the Kinetic Energy is at a maximum, then the
system will not have any Potential Energy.
Conservation of Mechanical Energy
ME = KE + PE
KEinitial + PEinitial = KEfinal + PEfinal
Example 4:
• A student with a mass of 55 kg goes down a
frictionless slide that is 3 meters high. What is the
student’s speed at the bottom of the slide?
KEinitial + PEinitial = KEfinal + PEfinal
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KEinitial = 0 because v is 0 at top of slide.
PEinitial = mgh
KEfinal = ½ mv2
PEfinal = 0 at bottom of slide.
– Therefore:
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PEinitial = KEfinal
mgh = ½ mv2
v = √2gh
V = (2)(9.81 m/s2)(3 m) = 7.67 m/s
Example 5:
•
A student with a mass of 55 kg goes goes down a
non-frictionless slide that is 3 meters high.
– Compared to a frictionless slide the
student’s speed will be:
a. the same.
b. less than.
c. more than.
•
Why?
•
Because energy is lost to the environment in
the form of heat (Q) due to friction.
Example 5 (cont.)
• Does this example reflect conservation
of mechanical energy?
• No, because of friction.
• Is the law of conservation of energy
violated?
– No, some of the “mechanical” energy is
lost to the environment in the form of heat.
Energy of Collisions
• While momentum is conserved in all
collisions, mechanical energy may
not.
– Elastic Collisions: Collisions where the
kinetic energy both before and after are
the same.
– Inelastic Collisions: Collisions where
the kinetic energy after a collision is
less than before.
• If energy is lost, where does it go?
• Thermal energy, sound.
Collisions
• Two types
– Elastic collisions – objects may deform but after the collision
end up unchanged
• Objects separate after the collision
• Example: Billiard balls
• Kinetic energy is conserved (no loss to internal energy or heat)
– Inelastic collisions – objects permanently deform and / or
stick together after collision
• Kinetic energy is transformed into internal energy or heat
• Examples: Spitballs, railroad cars, automobile accident
Example 4
• Cart A approaches cart B, which is initially at rest,
with an initial velocity of 30 m/s. After the collision,
cart A stops and cart B continues on with what
velocity? Cart A has a mass of 50 kg while cart B has
a mass of 100kg.
A
B
Diagram the Problem
A
B
Before Collision:
pA1 = mvA1
pB1 = mvB1 = 0
After Collision:
pA2 = mvA2 = 0
pB2 = mvB2
Solve the Problem
• pbefore = pafter
0
0
• mAvA1 + mBvB1 = mAvA2 + mBvB2
• mAvA1 = mBvB2
• (50 kg)(30 m/s) = (100 kg)(vB2)
• vB2 = 15 m/s
• Is kinetic energy conserved?
• KEi =? KEf
Solve the Problem
• mA = 50kg vA1 = 30m/s
• mB = 100kg vB2 = 15m/s
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Is kinetic energy conserved?
KEi =? KEf
KEi = Sum(½ mivi2)
KEf = Sum(½ mfvf2)
Example 5
Per 7
• Cart A approaches cart B, which is initially at
rest, with an initial velocity of 30 m/s. After the
collision, cart A and cart B continue on together
with what velocity? Cart A has a mass of 50 kg
while cart B has a mass of 100kg.
A
B
Diagram the Problem
A
B
Before Collision:
pA1 = mvA1
pB1 = mvB1 = 0
After Collision:
pA2 = mvA2
pB2 = mvB2
Note: Since the carts stick together after the collision, vA2 = vB2 = v2.
Solve the Problem
• pbefore = pafter
• mAvA1 + mBvB1 = mAvA2 + mBvB2
• mAvA1 = (mA + mB)v
0 2
• (50 kg)(30 m/s) = (50 kg + 100 kg)(v2)
• v2 = 10 m/s
• Is kinetic energy conserved?
• KEi =? KEf
Key Ideas
• Conservation of energy: Energy can be
converted from one form to another, but it is
always conserved.
• In inelastic collisions, some energy will be
lost as heat
• ET = KE + PE + Q
Key Ideas
• Gravitational Potential Energy is the energy
that an object has due to its vertical position
relative to the Earth’s surface.
• Elastic Potential Energy is the energy stored
in a spring or other elastic material.
• Hooke’s Law: The displacement of a spring
from its unstretched position is proportional
the force applied.
• Conservation of energy: Energy can be
converted from one form to another, but it is
always conserved.
Simple Harmonic Motion & Springs
• Simple Harmonic Motion:
– An oscillation around an equilibrium position in
which a restoring force is proportional the the
displacement.
– For a spring, the restoring force F = -kx.
• The spring is at equilibrium when it is at its relaxed
length.
• Otherwise, when in tension or compression, a restoring
force will exist.
Simple Harmonic Motion & Springs
• At maximum displacement (+ x):
– The Elastic Potential Energy will be at a maximum
– The force will be at a maximum.
– The acceleration will be at a maximum.
• At equilibrium (x = 0):
– The Elastic Potential Energy will be zero
– Velocity will be at a maximum.
– Kinetic Energy will be at a maximum
Harmonic Motion & The Pendulum
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Pendulum: Consists of a massive object called a
bob suspended by a string.
Like a spring, pendulums go through simple
harmonic motion as follows.
T = 2π√l/g
Where:
» T = period
» l = length of pendulum string
» g = acceleration of gravity
•
Note:
1.
2.
This formula is true for only small angles of θ.
The period of a pendulum is independent of its mass.
Conservation of ME & The Pendulum
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In a pendulum, Potential Energy is converted into
Kinetic Energy and vise-versa in a continuous
repeating pattern.
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PE = mgh
KE = ½ mv2
MET = PE + KE
MET = Constant
Note:
1.
2.
3.
Maximum kinetic energy is achieved at the lowest point of
the pendulum swing.
The maximum potential energy is achieved at the top of
the swing.
When PE is max, KE = 0, and when KE is max, PE = 0.