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Potential Energy and
Conservation of Energy
W08D
Conservative Forces
Definition: Conservative Force If the work done by a force in moving an object from an initial point to a final point is independent of the path (A or B),
then the force is called a conservative force which we denote by
c (path independent)
B c A
W โก โซ F โ
d r
G G
F c
G
Example: Gravitational Force
ย Consider the motion of an object under the influence
of a gravitational force near the surface of the earth
ย The work done by gravity depends only on the
change in the vertical position
W g = Fg ฮy = โmg ฮy
Concept Question: Energy
1. a
2. b
3. c
4. d
5. e
An object is dropped to the earth from a height of 10m. Which of the following sketches best represent the kinetic energy of the object as it approaches the earth (neglect friction)?
Change in Potential Energy
c
B
c A
ฮU โก โ โซ F โ
d r = โW
G G
Definition: Change in Potential Energy The change in potential energy of a body associated with a conservative force is the negative of the work done by the conservative force in moving the body along any path connecting the initial and the final positions.
F c
G
Work-Energy Theorem
The work done by the total force in moving an object from A to B is equal to the change in kinetic energy
When the only forces acting on an object are conservative forces
then the change in potential energy is
Therefore
0
total total 2 2 0
z f
W โก โซz F โ
d r = mvf โ mv โก ฮK
G G
ฮU = โWc = โW total
โฮU = ฮK
total F = F c
G G
Conservation of Energy for
Conservative Forces
When the only forces acting on an object are conservative
Definition: Mechanical Energy The mechanical energy is the sum of the kinetic and potential energies
Equivalently, the mechanical energy remains constant in time
ฮK + ฮU = 0
E (^) f^ mechanical^ = K (^) f + U (^) f = K (^) i + U (^) i = E (^) i^ mechanical
E mechanical^ โก K + U
Concept Question: Energy
Consider the following sketch of potential energy for a particle as a function of position. (There are no other non-conservative forces acting on the particle i.e. no dissipative forces or internal sources of energy.)
If a particle travels through the entire region of space shown in the diagram, at which point is the particle's velocity a maximum?
- a
- b
- c
- d
- e
Concept Question: Energy and
Choice of System
1. block
2. block + spring
3. block + spring + incline
4. block + spring + incline +
Earth
A block of mass m is attached to a relaxed spring on an inclined plane. The block is allowed to slide down the incline, and comes to rest. The coefficient of kinetic friction of the block on the incline is ฮผk. For which definition of the system is the change in total energy (after the block is released) zero?
Change in Potential Energy:
Inverse Square Gravity
Force:
Work done:
Potential Energy Change:
Zero Point:
Potential Energy Function
1 2
1 2 m , m 2 ห
Gm m r
F = โ r
G
0 0 0
(^12 2 1 21 ) 0
r (^) f rf (^) rf 1 1
r r r^ f
W d Gm m^ dr Gm m Gm m r r r r
โ โ^ โ^ โ = โ
= (^) โ โ (^) โ = = (^) โโ โ โโ โซ โซโ โ (^) โ โ F r
G (^) G
grav grav 1 2 0
1 1 f
U W Gm m r r
โ โ ฮ = โ = โ โโ โ โโ โ โ
U (^) grav ( r 0 = โ) = 0
1 2 grav ( ) =^
Gm m U r r
โ
Change in PE: Spring Force
Force:
Work done:
Potential Energy Change:
Zero Point:
Potential Energy Function
x
F = F ห i^ = โkxห i
G
( ) ( ) 0
2 2 spring 0
1 2
x x f f x x
W kx dx k x x
=
=
= (^) โซ โ = โ โ
spring spring (^2 02 )
1 2 f
ฮU = โW = k x โx
U spring ( x = 0) = 0
2 spring
U x = kx
Concept Question: Spring
In part (a) of the figure, a cart attached to a spring rests on a frictionless track at the position x (^) equilibrium and the spring is relaxed. In (b), the cart is pulled to the position xstart and released. It then oscillates about x (^) equilibrium. Which graph correctly represents the potential energy of the spring as a function of the position of the cart?
Force and Potential Energy
In one dimension, the potential difference is
Force is the derivative of the potential energy
Examples: (1) Spring Potential Energy:
(2) Gravitational Potential Energy:
U (x) = U (x 0 ) โ Fx A
B โซ dx
Fx = โ
dU dx 2 spring
1 ( ) 2
U x = kx 2 ,
1 x spring 2
dU d F kx kx dx dx
= โ = โ โ^ โ= โ โโ โโ 1 2 grav ( ) =^
Gm m U r r
โ 1 2 1 2 r gravity, 2
dU d Gm m Gm m F dr dr r r
= โ = โ โ^ โ โ= โ โโ โโ
Non-Conservative Forces
Work done on the object by the force depends
on the path taken by the object
Example: friction on an object moving on a
level surface
friction
friction friction 0
k
k
F N
W F x N x
Non-Conservative Forces
Definition: Non-conservative force Whenever the work done by a force in moving an object from an initial point to a final point depends on the path, then the force is called a non-conservative force.
Change in Energy for Conservative
and Non-conservative Forces
Total force:
Total work done is change in kinetic energy:
Energy Change:
( )
B B total total total total total c nc nc A A
W = (^) โซ F โ
d r = (^) โซ F + F โ
d r = โฮU +W = ฮK
G (^) G G G G
ฮK + ฮU total^ = Wnc
total total total F = F c (^) + F nc
G G G
