Lecture 3 work and energy 2015

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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

v d v m

ds dt

v

d m ds

F





s F Fs

cos 

 

2

2

1

mv

K 

F=const





s

2 1

2 2

2

1 2

1

mv mv

K

W    

The work done by a constant force F on the object when it moves a straight distance s is:

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:

2 1

2 2 2

1 2

1

2

1

v m v

m v

d v m ds

F

W

v

v









 









  

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

We define: Kinetic Energy:



F

ds (1)

(2)

The total work done on a particle is equal to the change in its kinetic energy

Work-Kinetic Energy Theorem

4.2 Power

P

mv mv

P

W

t

mv mv

W

t P W

2 1

2 2

2 1

2 2

2

1 2

1 2

1 2

1

const P

if





















time of

unit per

done

work Power

_ _

_

_

dt

ds F

dt

dW









 

1

t

t

Pdt dW

W

Conservative Nonconservative Force

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

Fgrv

ds

(1)

(2)

M

m

dr

r

2

r1

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

dr r

Mm G

ds F ds

F dW

dr

2

cos    

  

1 2

2

2

Mm G

r

Mm G

dr r

Mm G

W

r

r







 

Work done by the grav force F on object

m when it moves from (1) to (2)

+ 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 :

C r

Mm G

r

U( )   

2 2

11

/

10 67

.

Universal Graviattional Cconstant

If we choose U=  0 when r=  , we have C=0, If we choose U=0 on the

surface of Earth: C=GMm/R

U U

U

W  1  2   

We can write:

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

(1)

(2) (a)

(b)

U U

U ds

F ds

F W

b

c a

) 2 1 (

)

2 1 (

.

0

) (

. 

 

C

c ds F

(1)  (2)

F

ds

Mechanical Energy

• If an object is exerted by

Conservative Force Fc

and Nonconservative

Force Fnc,

• from the Work-Kinetic

Energy Theorem :

• Fc is conservative:

• From (1) and (2):

• Mechanical Energy:

E=K+U

nc

nc

F

F W U

K U

K

W U

U K

K

















) (

)

2 1

1 2

nc

F a





) 1

(

1

2 K WF c WF nc

K



) 2

(

2

1 U U

U

W

c

nc

F

W E

E

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)

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

nc

F

W E

E

Conservative Force

& Potential Energy

z

U F

y

U F

x

U F

dz z

U dy

y

U dx

x

U dU

dW

ve conservati is

F

dz F dy

F dx

F ds

F dW

k dz j

dy i

dx ds

k F j

F i

F

F

z y

x

z y

x c

z y

x

















































































;

;

) 2 (

_

_

) 1 (

















U grad

z

j y

i x



















Gradient Operator

Lực bảo tòan và thế năng

)

z

e y

e x

grad

























Dao động xung quanh vị trí cân bằng

bền

2 2

2

0

) (

2

1 )

( )

( )

k

x x

p e

x x

p e

p

x

x

x x

x x

e e













































) (

) (

/

e

e p

x

x x

k x

m

x x

k x

F

























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

vị trí cân bằng

2

)

.(

2

1 )

( )



0





































k

k m

x x

bền k>0

ko bền k<0

0 0

) (

2 2

2



































x e x p

e

x k

m k

x x











  

Pt chuyển động xq vị trí cân bằng

bền

Độ lệch khỏi vị trí cân bằng

Tần số dao động