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Chapter 6, Potential Energy and Energy Conservation
Chapter 6
POTENTIAL ENERGY AND ENERGY CONSERVATION
Introduction
Kinetic Energy is energy by virtue of motion. There exists other forms of energy:
potential (by virtue of position), chemical (chemical reactions give off or absorb
energy), thermal, etc...
VI-1 Gravitational Potential Energy
• A change in kinetic energy (energy by virtue of motion) is always due to forces
being exerted on a body.
• In the case of the gravitation force, a mass dropped from a higher point will reach
higher velocity (higher kinetic energy) than if it were dropped from a lower position.
There is more potential for work to be done depending on the altitude.
•
Energy by virtue of position is called Potential Energy
y1
The work done by the gravitation force on the
body is
W = Fs = -mg(y2-y1)
W = mgy1 - mgy2
U = mgy is called the potential energy
y2
• The change in potential energy:
∆U = (final value -initial value). Thus,
∆U = mgy2 - mgy1
Work done = W = - ∆U
• If g only acts, from the Work-Energy theorem, we have: W = ∆K, but W is only due to
g ⇒ ∆K = -∆U
1
mv12 + mgh1 = 12 mv 22 + mgh2
2
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Chapter 6, Potential Energy and Energy Conservation
•
(K + U) is called the total mechanical energy of the system
• In case where g only does work, we see that E = K + U = constant.
If E ↑ then U↓, also if E↓ then U↑
This is called Conservation of Mechanical Energy.
• The choice of the origin of coordinates (y = 0) does not affect the CHANGE (∆U) i n
gravitational potential energy.
Thus, U = 0 can be chosen anywhere without affecting the situation.
In case other forces act (i.e. do work):
Wtot = Wother+Wgrav = K2 - K1
•
Case of a curved path:
r
dx
r
w
r
dy
r
ds
Work done
by g along :
r
ds
r r
dWgrav = w.ds
r r
= −mg( dx + dy )
r
r
r
= −mgdx − mgdy = −mgdy
• That is: Wgrav = -mg (y2 - y1)
It is the same as a purely vertical path. The work done by the gravitation force is
unaffected by any horizontal motion that might occur.
VI-2 Elastic Potential Energy
• A body is called elastic if it returns to its original shape after being deformed. All
materials have elastic properties to a certain extent (see Chapter stress-strain). Typical
elastic medium is the helical spring.
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Chapter 6, Potential Energy and Energy Conservation
• If a force is applied to compress or stretch a spring attached to a body and then
released: the body will be set up in motion. All the stored energy has been converted to
kinetic energy. This is called elastic potential energy.
We saw that the work done to stretch from x1 to x2 is:
W = 12 kx 22 − 12 kx12 with x2 〉 x1 ,
This is the work done on the spring by an external force to stretch it.
• Now from the spring point of view: the force exerted by a stretched spring (x) on a
body is F = −kx . This is called the restoring force.
r
F
x
•
The work done by the spring on the body from x1 to x2 (x1 > x2) is:
x2
W=
•
∫ −kxdx =
1
2
kx12 − 12 kx 22
x1
The quantity
U = 12 kx 2
is called the elastic potential energy.
•
Graph of U(x)
X<0
Compressed
X>0
Stretched
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Chapter 6, Potential Energy and Energy Conservation
•
The work done on the block by the elastic force is equal to:
Wel = 12 kx12 − 12 kx 22 = U1 − U2 = − ∆U
•
We now apply the Work-Energy theorem. Only the elastic force does work:
Wtotal = ∆K = − ∆U
1
2
mv12 + 12 kx12 = 12 mv 22 + 12 kx 22
VI-3 Conservative and non-conservative forces
• Gravitation or elastic forces are conservative, i.e. total mechanical energy of the
system is conserved.
If the force acts in reverse, the energy is converted back into the other form.
• The work done by conservative forces has the following properties:
nIt can be represented by a potential energy function
oIt is reversible
p It does not depend on the path, but only on the starting and ending points.
• Non-conservative forces are said to be dissipative forces: some energy is
dissipated when they act.
• They cannot be represented by a potential function. e.g., kinetic friction, fluid
resistance, etc.
They depend on the path
• The effect of non-conservative forces can be taken into account by considering
changes in the internal energy of the body. This is associated with temperature
changes (see thermodynamics at the end of Semester 1).
It is possible to show that:
∆
∆
K
∆
U
∆
U
∆K
K +++∆
∆U
U +++∆
∆U
Uiniinnttt === 000
LLLaaaw
w
w ooofff cccooonnnssseeerrrvvvaaatttiiiooonnn ooofff eeennneeerrrgggyyy
VI-4 Relationship between Force and Potential Energy
•
Can we find the force from a specified potential function?
∆U
We had W = -∆U. For a small change: Fx∆x = -∆U ⇒ Fx = −
∆x
•
In the limit ∆x→0, then
Fx ( x ) = −
dU
dx
F x will always be directed in the direction opposite the direction of increasing
•
F tries to push the system towards regions of lower potential energy.
dU
.
dx
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Chapter 6, Potential Energy and Energy Conservation
•
In three dimensions, one could
r show that:
F = −gradU = −∇U
r
 ∂U r ∂U r ∂U r
F = −
i +
j+
k
∂y
∂z 
 ∂x
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