Download Physics I - Lecture 5 - Conservation of Energy

General Physics I
Lecture 5: Conservation of
Energy
Prof. WAN, Xin （万歆）
[email protected]
http://zimp.zju.edu.cn/~xinwan/
Outline
●
Revisit work-kinetic energy theorem
W =K f −K i=Δ K
●
●
Introduce potential energy (for conservative
forces)
W =U i−U f =−Δ U
Conservation of (mechanical) energy
E=K +U=const
●
Newton's
second law
Non-conservative forces
Work-Kinetic Energy Theorem
●
●
●
The net work done on a particle by the net force
acting on it is equal to the change in the kinetic energy
of the particle.
We can think of the kinetic energy as the work a
particle can do in coming to rest, or the amount of
energy stored in the particle.
What about other forms of energy?
Constant Acceleration
Set ti  0, t f  t , v xf  v xi  a x t
1
x f  xi  v xi  v xf t
2
1 2
 v xi t  a x t
2
2

v xf  v xi
2a x
2
v xf
vx
axt
2
a xt
2
v xi
0
t
Time to Forget
●
Work done
W =max (x f −xi )
●
●
Kinetic energy change
1
1
2
2
K f −K i= m v xf − mv xi
2
2
From the work-kinetic energy theorem
v 2xf −v2xi
x f −x i=
2 ax
Potential Energy
●
●
●
Roughly speaking, the energy associated with a
system of objects.
Consider a system of two particle-like objects. The
work done by the force acting on one of the objects
causes a transformation of energy between the
object's kinetic energy and other forms of the
system's energy.
Concrete examples:
− Elastic potential energy
− Gravitational potential energy
Elastic Potential Energy
●
The elastic potential energy of the system can be
thought of as the energy stored in the deformed
spring (one that is either compressed or stretched
from its equilibrium position).
xf
1
1
2
2
W s = ∫ (−kx) dx = k x i − k x f
2
2
x
i
= U i−U f
= −(U f −U i ) = −Δ U g
Elastic Potential Energy
●
The elastic potential
energy associated with
the system is defined
by
−
Zero: undeformed
−
Always positive in
a deformed spring
Gravitational Potential Energy
●
Gravitational potential energy is the potential
energy of the object–Earth system. The potential
energy is transformed into kinetic energy of the
system by the gravitational force.
−
The earth is much more massive than the object.
The earth can be modeled as stationary, and the
kinetic energy of the system can be represented
entirely by the kinetic energy of the lighter object.
−
The kinetic energy of the system is represented by
that of the object falling toward the Earth.
Gravitational Potential Energy
●
The work done by the
gravitational force
⃗
⃗ )⋅d
W g=(m g
=(−m g ŷ )⋅( y f − y i ) ŷ
=m g y i −m g y f
=U i−U f
=−(U f −U i )=−Δ U g
Gravitational Potential Energy
●
If a particle of mass m
is at a distance y above
the Earth’s surface, the
gravitational potential
energy of the particleEarth system is
Independent of the choice of
the origin.
Valid only for objects near the surface of the Earth.
Frictional Force
Although the details of friction are quite
complex at the atomic level, this force
ultimately involves an electrical
interaction between atoms or molecules.
Force of static friction
Force of kinetic friction
Frictional Force
●
Experimentally, we find that, to a good
approximation, both fs,max and fk are proportional
to the normal force acting on the object.
μ s: coefficient of
static friction
μ k: coefficient of
dynamic friction
Slow Down, Please!
A 1 500-kg car moving on a flat,
horizontal road which curves with
a radius 35.0 m and the coefficient
of static friction is 0.500, find the
maximum speed the car can make
the turn successfully.
Work Done by Frictional Force
⃗
W friction= f⃗k⋅d=−
f kd
Can we define a frictional potential like
we did in the case of gravitational force?
Example
●
Jumping from 2.00 m high
1
2
mgh= m v f −0
2
v f =√2 g h=6.26 m/ s
●
Sliding down
1
2
mgh− f̄k l= m v f −0
2
Depending on path!
Conserved or Nonconserved?
●
The path makes no difference when we consider
the work done by the gravitational force, but it
does make a difference when we consider the
energy loss due to frictional forces.
−
Gravitational force is conservative.
−
Frictional force is nonconservative.
Conservative Forces
●
●
A force is conservative if the work it does on a
particle moving between any two points is
independent of the path taken by the particle.
The work done by a conservative force on a
particle moving through any closed path is zero.
(A closed path is one in which the beginning and
end points are identical.)
A
B
Conservation of Mechanical Energy
●
The work done by a conservative force F to a
particle is
●
On the other hand, we have
●
Together, we have
Conservation of Mechanical Energy
●
The total mechanical energy of a system remains
constant in any isolated system of objects that
interact only through conservative forces.
−
No energy is added to or removed from the system.
−
No nonconservative forces are doing work within
the system.
−
Sum over the potential energy associated with each
conservative force.
Nonconservative Forces
●
●
●
Frictional force
Δ E=Δ K +Δ U =W friction
By the frictional force, mechanical energy is
transformed into internal energy (the kinetic energy
associated with the random motions of the atoms or
molecules and the potential energy associated with
the forces between the atoms or molecules).
We will revisit the issue during the discussion of the
first law of thermodynamics. Energy is conserved
after taking into account the internal energy.
Walter Lewin
Landing of the Actor
The largest tension
determines the maximum angle
without lifting the sandbag.
Conservative Force
●
●
Earlier, we learned how to determine the change
in potential energy of a system when we are given
the conservative force.
Now we show how to find the force if the potential
energy of the system is known.
Derivative of Potential Energy
●
●
Any conservative force acting on an object within
a system equals the negative derivative of the
potential energy of the system with respect to x.
Deformed spring
dU s
d 1
2
F s =−
=−
k x =−kx
dx
dx 2
(
●
)
Gravitation
dU g
d(mgy )
F g=−
=−
=−mg
dy
dy
Vector Form
●
In the language of vector calculus, F equals the
negative of the gradient of the scalar quantity U(x,
y, z). In three dimensions,
You will learn in the case of electric field and electric potential
Equilibrium of a System
In general, positions of
stable equilibrium
correspond to points for
which U(x) is a minimum.
dU
=0
dx
2
d U
>0
2
dx
Unstable Equilibrium
In general, positions of
unstable equilibrium
correspond to points for
which U(x) is a maximum.
dU
=0
dx
2
d U
<0
2
dx
Lennard–Jones Potential
●
The potential energy associated with the force
between two neutral atoms in a molecule can be
modeled by the Lennard–Jones potential energy
function:
Mass-Energy Equivalence
●
●
●
Energy can never be created or destroyed. Energy
may be transformed from one form to another,
but the total energy of an isolated system is always
constant.
Conservation of mass states that in any physical or
chemical process, mass is neither created nor
destroyed.
We will learn in the theory of
special relativity mass and energy
are not separately conserved.
Quantization of Energy