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Chapter 8
Lecture 15:
Conservation of Energy: II
Conservation of Energy

Energy is conserved


This means that energy cannot be created nor
destroyed
If the total amount of energy in a system changes,
it can only be due to the fact that energy has
crossed the boundary of the system by some
method of energy transfer
8.5: Conservation of Mechanical Energy
Principle of conservation of energy:
In an isolated system where only conservative forces cause energy changes, the
kinetic energy and potential energy can change, but their sum, the mechanical
energy Emec of the system, cannot change.
The mechanical energy Emec of a system is the sum of its potential energy U and the
kinetic energy K of the objects within it:
With
We have:
and
Example – Free Fall

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Determine the speed of
the ball at y above the
ground
Conceptualize

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Use energy instead of
motion
Categorize

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System is isolated
Only force is gravitational
which is conservative
Example – Free Fall, cont

Analyze

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Apply Conservation of Energy
Kf + Ugf = Ki + Ugi

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Ki = 0, the ball is dropped
Solving for vf
Finalize
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The equation for vf is consistent with the results
obtained from kinematics
Example – Spring Loaded Gun

Conceptualize

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The projectile starts from
rest
Speeds up as the spring
pushes against it
As it leaves the gun, gravity
slows it down
Categorize

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System is projectile, gun,
and Earth
Model as a system with no
nonconservative forces
acting
Example – Spring Gun, cont
Kinetic Friction



Kinetic friction can be
modeled as the
interaction between
identical teeth
The frictional force is
spread out over the
entire contact surface
The displacement of
the point of application
of the frictional force is
not calculable
Work – Kinetic Energy With
Friction

In general, if friction is acting in a system:


DK = SWother forces -ƒkd
This is a modified form of the work – kinetic
energy theorem

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Use this form when friction acts on an object
If friction is zero, this equation becomes the same
as Conservation of Mechanical Energy
Example – Block on Rough
Surface


The block is pulled by a
constant force over a
rough horizontal
surface
Conceptualize


The rough surface
applies a friction force on
the block
The friction force is in the
direction opposite to the
applied force
Example – Rough Surface cont
Example – Ramp with Friction

Problem: the crate slides
down the rough ramp


Conceptualize


Find speed at bottom
Energy considerations
Categorize


Identify the crate, the
surface, and the Earth as
the system
Isolated system with
nonconservative force
acting
Example – Ramp, cont

Analyze





Let the bottom of the ramp be y = 0
At the top: Ei = Ki + Ugi = 0 + mgyi
At the bottom: Ef = Kf + Ugf = ½ m vf2 + mgyf
Then DEmech = Ef – Ei = -ƒk d
Solve for vf
Sample problem: energy, friction, spring, and tamales
System: The package–spring–floor–wall system
includes all these forces and energy transfers in
one isolated system. From conservation of
energy,
Forces: The normal force on the package from the
floor does no work on the package. For the same
reason, the gravitational force on the package does no
work. As the spring is compressed, a spring force does
work on the package. The spring force also pushes
against a rigid wall. There is friction between the
package and the floor, and the sliding of the package
across the floor increases their thermal energies.
Collision and Impulse
In this case, the collision is brief, and the ball
experiences a force that is great enough to slow,
stop, or even reverse its motion.
The figure depicts the collision at one instant.
The ball experiences a force F(t) that varies
during the collision and changes the linear
momentum of the ball.
Linear Momentum

The linear momentum of a particle, or an
object that can be modeled as a particle, of
mass m moving with a velocity v is defined to
be the product of the mass and velocity:
 p  mv

The terms momentum and linear momentum will be
used interchangeably in the text
Linear Momentum, cont

Linear momentum is a vector quantity

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Its direction is the same as the direction of the
velocity
The dimensions of momentum are ML/T
The SI units of momentum are kg · m / s
Momentum can be expressed in component
form:

px = m v x
py = m vy
pz = m vz
Newton’s law and Momentum

Newton’s Second Law can be used to relate
the momentum of a particle to the resultant
force acting on it
dv d  mv  dp
SF  ma  m


dt
dt
dt
with constant mass
Conservation of Linear
Momentum
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Whenever two or more particles in an
isolated system interact, the total momentum
of the system remains constant
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The momentum of the system is conserved, not
necessarily the momentum of an individual
particle
This also tells us that the total momentum of an
isolated system equals its initial momentum
Conservation of Momentum, 2
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Conservation of momentum can be expressed
mathematically in various ways

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In component form, the total momenta in each
direction are independently conserved

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ptotal = p1 + p2 = constant
p1i + p2i = p1f + p2f
pix = pfx
piy = pfy
piz = pfz
Conservation of momentum can be applied to
systems with any number of particles
This law is the mathematical representation of the
momentum version of the isolated system model
Conservation of Momentum,
Archer Example
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The archer is standing
on a frictionless surface
(ice)
Approaches:

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Newton’s Second Law –
no, no information about
F or a
Energy approach – no,
no information about
work or energy
Momentum – yes
Archer Example, 2

Conceptualize

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Categorize
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The arrow is fired one way and the archer recoils in the
opposite direction
Momentum
 Let the system be the archer with bow (particle 1) and the
arrow (particle 2)
 There are no external forces in the x-direction, so it is
isolated in terms of momentum in the x-direction
Analyze

Total momentum before releasing the arrow is 0
Archer Example, 3

Analyze, cont.
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The total momentum after releasing the arrow is
p1f  p2f  0
Finalize
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The final velocity of the archer is negative
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Indicates he moves in a direction opposite the arrow
Archer has much higher mass than arrow, so velocity
is much lower