Ch.12 Energy II: Potential energy Ch. 12 Energy II: Potential energy

Ch.12 Energy II:Potential energy

Ch. 12 Energy II:Potential energy


12-1 Conservative forces( 保守力 )

Potential energy? It is defined only for a certain class of forces called conservative forces.

Do spring force, gravitational force, and frictional force et al. belong to conservative forces?

Kinetic energy Velocity

What are conservative forces?


1. The spring force Fig 12-1

x

0

0

d

d

-d

(a)

(b)

(c)

(d)

(e))(

2

1 22ifs xxkW

ox

Relaxed length

Fig 11-13

The total work done by the spin force is zero in the process from (a) to (e) (round trip).

0

0


The total work done by the gravity is zero during the round trip.

2. The force of gravity

If the gravitational force is not constant, is there still such behavior of the work?

See See 动画库动画库 // 力学夹力学夹 /2-04/2-04功功的计算举例的计算举例 (2)(2)


3. The frictional force

The total work done by frictional force is not zero in a round trip.

See See 动画库动画库 // 力学夹力学夹 /2-04/2-04功功的计算举例的计算举例 (1)(1)


Definition of conservative force:

One particle exerted by a force moves around a closed path and returns to its starting point.

If the total work done by the force during the round trip is zero, we call the force ‘a conservative force’ ， such as spring force and gravity.

If not, the force is a nonconservative one.


Two Mathematical statements:

a

a

b

b1

2

2

1

Fig 12-4

(a)

(b)

If is a conservative force, we have:

F

0WW ba.2ab.1

0sdFsdFa

b

b

a

(12-1)Path1 Path2

Statement 1

sdFsdFsdFb

a

a

b

b

aPath1 Path2Path2

(12-3)

0 sdF

Statement 2

sdFsdFb

a

b

aPath1 Path2


To every action, there is an equal and opposite To every action, there is an equal and opposite reaction.reaction.

Newton’s third lawNewton’s third law

Note:

(1) Both the action and reaction forces belong to the system.(2) The total work done by action and reaction forces is independent of the reference frame chosen (even in non-inertial frame).

Prove point (2):


1f

2f

Ｏ

1r

2r

z

y

x

m1 m2221121 rdfrdfWWW

SIn S frame:

In S’ frame with velocity ofrelative to S frame:

ssv '

'rdf'rdf'W'WW' 221121

)()( 2211 dtvrdfdtvrdf s'ss's

dtv)ff(rdfrdf s's

212211 21 ff

2211 rdfrdf

W


12-2 Potential energy1.Definition

When work is done in a system (such as ball and earth) by a conservative force, the configuration of its parts changes, and so the potential energy changes from its initial value to its final value . We define the change in potential energy associated with the conservative force as:

iU fU

sdFWUUΔU if

(12-4)


2. The potential energy of gravityFor the ball-Earth system, we take upward direction to be y positive direction

)()()()( 12

2

112 yymgdymgyUyUUy

y

The physically important quantity is ,ΔU not or .)( 2yU )( 1yU

0)0( 1 yU

mgyyU )( (12-9)

If We set

We have

(the reference zero point of U is at O)

y2

y1

y

sdFWUUΔU if

mg

)()()( 1212 yymgyUyU , dependent on )( 1yU


3. The potential energy of spring force

00 u

x

kx)dx(FdxU(x)0

0

2

2

1)( kxxU

o x

Relaxed length

Fig 11-13

When the spring is in its relaxed state, and we can declare the potential energy of the system to be zero ( )

(12-8)

The reference zero point of potential is at x=0.

2

1


i.The physically important quantity is . Not or .

Notes:

01 xx UUΔU

1xU 0xU

iii.Potential energy belongs to the system (Such as ball-Earth) and not of any of the individual objects within the system.

ii.We are free to choose the reference point at any convenient location for the potential energy.


iv. The inverse of Eq(12-4) allows us to calculate the force from the potential energy (12-7)

dx

xdUxFx

)()(

x

x FdxUUΔU00

Eq(12-7) gives us another way of looking at the potential energy:

“The potential energy is a function of position whose negative derivative gives the force”

Eq(12-4)

mgyyU )( yFmgdy

ydU

)(

2

2

1)( kxxU xFkxkx

dx

d

dx

dU )

2

1( 2


An elevator cab of mass m=920 Kg moves from street level to the top of the World Trade Center in New York, a height of h=412 m above ground. What is the change in the gravitational potential energy of the cab-Earth system?

Sample problem 12-1

.107.34128.9920 6 JmghymgU


12-3 Conservative of mechanical energy

dxxFWUUΔU if )(

KΔU K

mvmvW ifnetF

22

2

1

2

1

From the definition of potential energy, we have:

K

0 KU (12-14)

0)()( ifif KKUU

ffii UKUK (12-15)

Mechanical energy

When can Eq. (12-15) be satisfied?

is a conservative forceF


Eq(12-15) is the mathematical statement of the law of conservation of mechanical energy:

“In a system in which only conservative forces do work, the total mechanical energy of the system remains constant.”

Such as the systems of: Ball-Earth system; Block-spring system on frictionless table.


How to write the formula of conservation of mechanical energy for:

Ball-Earth system / m+M system (M>>m) ?

No other forces exerted in the system.

.2

1

2

1)( 22 constMVmvRrU

m and v are the mass and speed for Ball, respectively.M and V are the mass and speed for Earth, respectively.

.2

1 2 constmvmgh or

Which one is correct or both correct?(1)If of M is zero, M is an inertial frame. Take M as our reference frame. Two eqs. are equivalent.

a

(2) If of M is not zero, CM of the system is an inertial frame. Since M>>m, the position of CM is very near M. In CM frame, V~0, R~0. So two eqs. are equivalent.

a


Sample problem 12-5

Using conservation of mechanical

energy, analyze the Atwood’s

machine (sample problem 5-5) to

find v and a of the blocks after they

have moved a distance y from rest.

Solution: We take the two blocks

plus the Earth as our system. For

simplicity, we assume that both

o

y

1m 2m


blocks start from rest at the same level, which we define as y=0, the reference point for gravitational potential energy.

Thus

Solving for the speed v, we obtain

0)2

1()

2

1( 2

221

21 gymvmgymvmUK ff

0,0 ii UK

gymm

mmv

21

122

gmm

mm

dt

dvay

21

12

,0 ii UK


When the climber goes down, she must transfer potential energy to other kinds of energy, such as thermal energy.


12-4 Energy conservation in rotational motion

We still restrict our analysis to the case in which the rotational axis remains in

the same direction in space as

the object moves. Fig 12-7 shows an arbitrary body of mass M .

Fig12-7

0

y

x

cmr

nr

'nr

c

p


1. Relative to o, the kinetic energy of is ,

and total kinetic energy of the body is

(12-16)

From Fig12-7,we see that Then:

nm2

2

1nnvm

2

1 2

1nn

N

n

vmK

'ncmn rrr

'ncmn vvv

)2(2

1

)()(2

1

2

1

'2'2

1

''2

ncmncmn

N

n

ncmncmnnn

vvvvm

vvvvmvmK

(12-17)


In Eq(12-17):

(1) ( )

(2)

(3)

22

2

1

2

1cmcmn Mvvm Mmn

22'2'

2

1)(

2

1

2

1 cmnnnn Irmvm

0)2(2

1 ''

nncmncmn vmvvvm

)'(

cmnn vvv


Thus (12-18)

Eq(12-18) indicate that the total kinetic energy of the moving object consists of two terms, the pure translational of Cm, and the pure rotation about Cm. The two terms are quite independent.2.Rolling without slipping For this case,

22

2

1

2

1 cmcm IMvK

Rvcm 22222

2

1

2

1)(

2

1

2

1 cmcm

cmcm IRMR

vIMvK (12-19)


3.When an object rolls without slipping, there may be a frictional force exerted at the instantaneous point of contact between the object and the surface on which it rolls. However, this frictional force does no work on the object.


Sample problem 12-8

Using energy conservation

find the final speed of the

rolling cylinder in Fig 9-32

when it reaches the bottom

of the plane.

h

mg

fN

c

Fig 9-32


Solution: For our system we take the cylinder and

the Earth. The friction does no work and so it cannot change the mechanical energy. ,

Setting

with

mghUKE iii 0

0)(2

1

2

1 22 R

vIMvUKE cmcmcmfff

ghvcm 3

4

2

2

1MRI cm

mghR

vIMv mcmcm 22 )(

2

1

2

1fi EE


12-5 One-dimensional conservation system: the complete solution1. How to read the curve of potential energy? If only conservative forces do work in the

system, we have , (E is constant) (12-20)UKE

EmvxU x 2

2

1)(

(12-21))]([2

xUEm

vx )(xUE


dx

dUF

6x 5x

(a)

4x3x 2x1x

0E

3E

4E

F

1E

(b)

2E

0x

Fig12-8

U(x)

)]([2

xUEm

vx

EU

K=E-U

x

x


Sample problem 12-10

The potential energy function for the force between

two atoms in a diatomic molecule can be expressed approximately as follows

where a and b are positive constant and x is the distance between atoms.(a)Find the equilibrium separation between the atoms, (b) the force between the atoms, and (c) the minimum energy necessary to break the molecule apart.

612)(

x

b

x

axU


Solution: (a) Fig12-9 shows u(x) as a function of x.

Equilibrium occurs at the coordinate

, where is a

minimum.

that is

mx)( mxU

0)( mxxdx

dU

0612713

mm x

b

x

a 6

1

)2(b

axm

x

x0

0

U

(a)

(b)

mx

xF

Fig (12-9)


(b)

(c) The minimum energy needed to break up the

molecule into separate atoms is called the

dissociation energy , .

713

612)(

x

b

x

a

dx

dUxFx

0)( ExUE mddE

a

bEd 4

2

612)(mm

mdx

b

x

axUE ))

2(( 6

1

b

axm


2. General solution for x(t)From Eq(12-21) ,with we have (12-22))]()[/2( xUEm

dxdt

dt

dxvx

)]()[/2(0 xUEm

dxt

x

x

Suppose , , we have (12-23)After carring out the integration of Eq(12-23), we would obtain t=t(x), or x=x(t).

00)( xx t xx t )(

)]([2

xUEm

vx


In one dimension, the magnitude of the gravitational force of attraction between two particles of mass m1 and mass m2 is given by

where G is a constant and x is the distance between the particles. (a) What is the potential energy function U(x)? Assume that U(x)0, as x ∞. (b) How much work is required to increase the separation of the particles form x=x1 to x=x1+d?

221

x

mmGFx

Exercise:

Documents

Ch.12 Energy II: Potential energy Ch. 12 Energy II: Potential energy