Download Lecture 08 - Eunil Won

PHYS151
Lecture 08
Ch 08 Potential Energy and Conservation of Energy
Eunil Won
Korea University
Fundamentals of Physics by Eunil Won, Korea University
Potential Energy
Potential energy is the energy that can be associated with the configuration
These two states has two
different potential energy
ex) gravitational potential energy
elastic potential energy
Work and Potential Energy
The change ΔU in gravitational potential energy is defined to equal
the negative of the work done on the tomato by gravitational force
Fundamentals of Physics by Eunil Won, Korea University
∆U = −W
Conservative and Nonconservative Forces
Conservative force:
Work done by a force to change configuration is same but negative
to the work done to restore the configuration
ex) gravitational force
Nonconservative force:
A force that is not conservative
ex) kinetic frictional energy (thermal energy cannot be transferred
back to kinetic energy)
Fundamentals of Physics by Eunil Won, Korea University
Path Independence of Conservative Forces
The net work done by a
conservative force on a particle
moving around every closed
path is zero
1
mv02
2
The work done by a conservative
force on a particle moving
between two points does not
depend on the path taken by the
particle
Wab,1 + Wba,2 = 0
Wab,1 = −Wba,2 = Wab,2
Fundamentals of Physics by Eunil Won, Korea University
1
mv02
2
Potential Energy Values
Consider a particle with a
conservative force
W =
!
xf
xi
Gravitational Potential Energy:
∆U = −
!
F (x)dx or ∆U = −
yf
(−mg)dy = mg
!
yf
dy = mg(yf − yi )
(if we take Ui = yi = 0)
U (y) = mgy
Elastic Potential Energy:
∆U = −
!
xf
(−kx)dx = k
!
xf
xi
xi
1 2
U (x) = kx
2
Fundamentals of Physics by Eunil Won, Korea University
xf
xi
yi
yi
!
1 2 1 2
x dx = kxf − kxi
2
2
(if we take Ui = xi = 0)
F (x)dx
Conservation of Mechanical Energy
The mechanical Energy Emec
Emec = K + U
When a conservative force does work W on an object, it transfers energy between K and U
∆K = W = −∆U
from the top, we find that
K2 − K1 = −(U2 − U1 )
and becomes
K2 + U2 = K1 + U1
Principle of conservation of mechanical energy:
∆Emec = ∆K + ∆U = 0
Fundamentals of Physics by Eunil Won, Korea University
Conservation of
Mechanical
Energy
ex) a pendulum
K + U is constant
all the time
Fundamentals of Physics by Eunil Won, Korea University
Potential Energy Curve
We had this before: ∆U = −
!
xf
F (x)dx
xi
and it becomes at the
differential limit:
dU (x)
F (x) = −
dx
(obtaining the force from the potential)
ex) elastic potential
1 2
U (x) = kx
2
F (x) = −kx
U (y) = mgy
F = −mg
gravitational potential
Fundamentals of Physics by Eunil Won, Korea University
Potential Energy Curve
Potential Energy Curve
neutral equilibrium
(U = 4J)
unstable equilibrium
(U = 3J)
Fundamentals of Physics by Eunil Won, Korea University
stable equilibrium
(U = 1J)
Work Done on a System by an External Force
Definition of work with external force:
Work is energy transferred to or
from a system by means of an
external force acting on that system
No friction involved:
W = ∆Emec
Work done on the system
is equal to the change in
the mechanical energy
Fundamentals of Physics by Eunil Won, Korea University
Work Done on a System by an External Force
Friction involved:
A constant force F pulls a block, increasing
the block’s velocity from v0 to v
From Newton’s 2nd law, we write
F − fk = ma
acceleration is constant as forces are constant,
so we can use the following equation
2
v =
v02
+ 2ad
and,
1 2
(v − v02 )
a=
2d
1
1
2
F d = mv − mv02 + fk d
2
2
it becomes
(if we include the vertical motion as well)
F d = ∆K + fk d
F d = ∆Emec + fk d
(increase in thermal energy by sliding)
Fundamentals of Physics by Eunil Won, Korea University
∆Eth
Conservation of Energy
The total energy E of a system can change only by amounts of energy that are
transferred to or from the system
W = ∆E = ∆Emec + ∆Eth + ∆Eint
(internal Energy)
Isolated system: there can be no energy transfers to or from the isolated system
The total energy E of an isolated system cannot change
∆Emec + ∆Eth + ∆Eint = 0
Fundamentals of Physics by Eunil Won, Korea University
Summary
Potential energy is the energy that can be associated with the configuration
Work and Potential Energy
∆U = −W
Principle of conservation of mechanical energy:
∆Emec = ∆K + ∆U = 0
Force and potential energy:
dU (x)
F (x) = −
dx
Conservation of energy:
W = ∆E = ∆Emec + ∆Eth + ∆Eint
Fundamentals of Physics by Eunil Won, Korea University