Lecture 3 work and energy 2015

Lecture 4
WORK and ENERGY


OUTLINE
•
•
•
•

Work and Kinetic Energy
The Work-Kinetic Energy Theorem
Power
Conservative Force-Nonconservative
Force
• Potential Energy
• Mechanical Energy
• Conservation of Mechanical Energy


4.1 Work and Kinetic Energy
The work done by a constant force F on the object when it
moves a straight distance s is:

F=const


W  Fs cos  F .s




s

In general case, the work is not constant, the path is a curve.

The work done by force F when the object moves a very small
displacement ds (we can consider F constant and ds a straigh

linet:


F


ds

(1)

(2)

 
dv
dW  F .ds  m .ds  mv dv
dt

The work done by force F when the object moves from position (1) to (2) is:

v2


  1  1 

W   F .ds   mv dv  mv22  mv12

2
2
1
v1
2

We define: Kinetic Energy:

K

1 2
mv
2

The total work done on a particle is equal to the change in its kinetic energy
1
1
W  K  mv22  mv12
Work-Kinetic Energy Theorem
2
2


4.2 Power
work _ done
Power 
per _ unit _ of _ time


dW F .ds  
P

 F .v
dt
dt
t2

W   dW   Pdt
t1

if P  const  W  Pt
1 2 1 2
W  mv2  mv1
2
2
1 2 1 2
mv2  mv1
W 2
2
t 

P
P


Conservative
Nonconservative Force
1. Definition: A force is Conservative if the

work done by the force is independent on
the path, it is dependent only on the
initial and final position.
- Gravity and spring force are
conservative forces,while kinetic friction
is not.
2. Work done by Gravitation Force:


(1)
r1

Fgrv

dr

m
ds

Work done by the grav. force F on object
m when it moves a displacement ds:


Mm
dW  F .ds   F .ds
.
cos




G
dr



2
r
dr

(2)Work done by the grav. force F on object
m when it moves from (1) to (2)

r

r2

2

M

G  6.67 10 11 N .m2 / kg 2

Mm
Mm
Mm
W    G 2 dr  G
G
r
r2
r1

r1

Universal Graviattional Cconstant
+ Work done is independent of the path, but of the initial and final position
gravitation force is conservative.
+ we define a scalar quantity called gravitational potential energy of two object
sseparated by a distance r :
Mm

U ( r )  G

r

C

If we choose U=0 when r=, we have C=0, If we choose U=0 on the
surface of Earth: C=GMm/R
We can write: W  U1  U 2  U


Conservative Forces
1. The work done by a conservative force on a particle moving
between any two points is independent of the path taken by
the particle.
2. 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.)
3. the work Wc done by a conservative force on an object as
the object moves from one position to another is equal to the
initial value of the potential energy minus the final value.

(2)
(a)


W12   Fc. .ds 
(1a 2 )

F
(b)
(1)

ds

(1)(2)


 Fc..ds  U1  U 2  U

(1b 2 )


W   Fc. .ds  0
(C )


Mechanical Energy
  
ma  Fc  Fnc

• If an object is exerted by

Conservative Force Fc
and Nonconservative
Force Fnc,
• from the Work-Kinetic
Energy Theorem :
• Fc is conservative:

WFc  U  U1  U 2 (2)

• From (1) and (2):

K 2  K1  U1  U 2  WFnc

K  K 2  K1  WFc  WFnc (1)

( K 2  U 2 )  ( K1  U1 )  WFnc
• Mechanical Energy:
E=K+U

E  E2  E1  WFnc


Conservation of Mechanical
Energy
The change in Mechnaical energy of an
object is equal to the work done by
nonconservative force on the object as it
takes a path form position (1) to (2)

E  E2  E1  WFnc

If Fnc= 0 or WFnc =0 =>E=0: E=const
Conservation of Mechanical Energy


Conservative Force
& Potential Energy







F  Fx i  Fy j  Fz k ds  dxi  dyj  dzk

dW  Fc .ds  Fx dx  Fy dy  Fz dz
(1)
F _ is _ conservative
 U
U
U 
dW  dU  
dx 
dy 
dz (2)
y
z 
 x
U
U

U
Fx  
; Fy  
; Fz  
x
y
z


F   gradU

Gradient Operator

     
grad  i 
j k
x
y
z


Lực bảo tòan và thế năng

 p   p   p 
F   grad p  (
ex 
ey 
ez )
x
y

z


Dao động xung quanh vị trí cân bằng
bền
Khai triển Taylor hàm thế năng xung quanh vị trí cân bằng:

 p

1  p
 p ( x)   p ( xe ) 
( x  xe )  . 2
( x  xe ) 2
x x  x
2 x x  x
e




e
0

vị trí cân
bằng

1
 p ( x)   p ( xe )  k .( x  xe ) 2
2

Fx   p / x  k ( x  xe )
mx  k ( x  xe )

2

k

bền k>0
ko bền k<0

  x  xe    x
m  k
k  0     2  0


Pt chuyển động xq vị trí cân bằng
bền
   2  0
Độ lệch khỏi vị trí cân bằng

  x  xe

Tần số dao động

k

m
  2 p 
k  2 
0

 x 

 ( x  xe )