chapter 7 kinetic energy and work. 7.3 kinetic energy kinetic energy k is energy associated with the...

Chapter 7 Kinetic energy and work

7.3 Kinetic energy

Kinetic energy K is energy associated with the state of motion of an object. The faster the object moves, the greater is its kinetic energy.

For an object of mass m whose speed v is well below the speed of light,

The SI unit of kinetic energy (and every other type of energy) is the joule (J),

1 joule = 1 J = 1 kgm2/s2.

7.4: Work

Work W is energy transferred to or from an object by means of a force acting on the

object.

Energy transferred to the object is positive work, and energy transferred from the

object is negative work.

7.5: Work and kinetic energy

To calculate the work a force F does on an object as the object moves through some displacement d, we use only the force component along the object’s displacement. The force component perpendicular to the displacement direction does zero work.

For a constant force F, the work done W is:

A constant force directed at angle to the displacement (in the x-direction) of a bead does work on the bead. The only component of force taken into account here is the x-component.

When two or more forces act on an object, the net work done on the object is the sum of the works done by the individual forces.

net by individual forcesW W

7.6: Work done by gravitational force

(a) An applied force lifts an object. The object’s displacement makes an angle =180° with the gravitational force on the object. The applied force does positive work on the object.

(b) An applied force lowers an object. The displacement of the object makes an angle with the gravitational force .The applied force does negative work on the object.

The theorem says that the change in kinetic energy of a particle is the net work done on the particle.

7.5: Work and kinetic energy

Work-kinetic energy theorem

It holds for both positive and negative work: If the net work done on a particle is positive, then the particle’s kinetic energy increases by the amount of the work, and the converse is also true.

ΣW = Kf – Ki = ΔK

Workdone on an accelerating elevator cab

7.8: Work done by a general variable force

A. One-dimensional force, graphical analysis:

•We can divide the area under the curve of F(x) into a number of narrow strips of width x.

•We choose x small enough to permit us to take the force F(x) as being reasonably constant over that interval.

•We let Fj,avg be the average value of F(x) within the jth interval.

•The work done by the force in the jth interval is approximately

I

•Wj is then equal to the area of the jth rectangular, shaded strip.

xFWW

xFW

avg,jj

avg,jj


A. One-dimensional force, calculus analysis:

We can make the approximation better by reducing the strip width x and using more strips (Fig. c). In the limit, the strip width approaches zero, the number of strips then becomes infinitely large and we have, as an exact result,

f

i

x

xavg,j0xdx)x(FxFlimW

Hooke’s Law

• When x is positive (spring is stretched), F is negative

• When x is 0 (at the equilibrium position), F is 0

• When x is negative (spring is compressed), F is positive

Slide 10

Fs = - kx

7.7: Work done by a spring force

Work Done by a Spring• Identify the block as the system

• Calculate the work as the block moves from xi = - xmax to xf = 0

• The total work done as the block moves from

–xmax to xmax is zero

Slide 11

max

0 2max

1

2

f

i

x

s xx xW F dx kx dx kx

xi xf

?f

i

x

s xxW F dx 2 2

i f

1 1 - 2 2

f

i

x

s xxW F dx kx kx

7.7: Work done by a spring force

Hooke’s Law: To a good approximation for many springs, the force from a spring is proportional to the displacement of the free end from its position when the spring is in the relaxed state. The spring force is given by

The minus sign indicates that the direction of the spring force is always opposite the direction of the displacement of the spring’s free end. The constant k is called the spring constant (or force constant) and is a measure of the stiffness of the spring.

The net work Ws done by a spring, when it has a distortion from xi to xf , is:

kxFs

Work Ws is positive if the block ends up closer to the relaxed position (x =0) than it was initially. It is negative if the block ends up farther away from x =0. It is zero if the block ends

up at the same distance from x= 0.


B. Three dimensional force:

If

where Fx is the x-components of F and so on,

and

where dx is the x-component of the displacement vector dr and so on,

then

Finally,

Θ

A boy of mass m is initially seated on the top of a hemispherical ice mound of radius R . He begins to slide down the ice, with a negligible initial speed. Approximate the ice as being frictionless.

(1)Calculate the work done by the gravitational force on the boy when the boy has moved through an angle θ with the vertical.(2)Calculate the work done by by the normal force.(3)What are the tangential and normal components of acceleration of the boy at an angle θ?

Work Done by Gravitational Force

• Generalizing gravitational potential energy uses Newton’s Law of Universal Gravitation:

• Calculate the work as the object moves from ri to rf

Slide 15

2ˆF rE

g

GM m

r

r r

G G 2r rˆr ? r r

f f

i i

EGM mW F d d

r

��

7.8: Work kinetic energy theorem with a variable force

A particle of mass m is moving along an x axis and acted on by a net force F(x) that is directed along that axis.

The work done on the particle by this force as the particle moves from position xi to position xf is :

But,

Therefore,

Power• The time rate of energy transfer is called power

• The average power is given by

Slide 17

avg

WP

t

Instantaneous Power• The instantaneous power is the limiting value

of the average power as t approaches zero

• This can also be written as

Slide 18

0lim

t

W dWP

t dt

rF F v

dW dP

dt dt

The SI unit of power is the joule per second, or Watt (W). In the British system, the unit of power is the footpoundper second. Often the horsepower is used. 1N=1 kg-m/s2

= 0.225 lb1 m = 3.281 ft

7.9: Power

Chapter 8

Potential energy and conservation of energy

8.1 Potential energy

Technically, potential energy is energy that can be associated with the configuration (arrangement) of a system of objects that exert forces on one another.

Some forms of potential energy: 1.Gravitational Potential Energy, 2.Elastic Potential Energy

Conservative Forces

• The work done by a conservative force 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

Slide 22

8.3 Path Independence of Conservative Forces

The net work done by a conservative force on a particle moving around any closed path is zero.

If the work done from a to b along path 1 as Wab,1 and the work done from b back to a along path 2 as Wba,2. If the force is conservative, then the net work done during the round trip must be zero

If the force is conservative,

Nonconservative Force, Example

• Friction is an example of a nonconservative force– The work done

depends on the path

– The red path will take more work than the blue path

Slide 24

• This system consists of Earth and a book

• Do work on the system by lifting the book through y

• The work done is mgyb - mgya

Slide 25

U

U U U W

W W

U W

b

b

a

a

重

外力

外力重

力

力

(移轉過程中不 )加速

F 外力

Gravitational Potential Energy

Determining Potential Energy

Ub

Ua

= mgyb - mgya

W

W W

b a

b a

外力

外力重力

重力

定義 U

U

U

U W

U

F 外力

Gravitational Potential Energy

Ub

Ua

= mgyb - mgya

0

W

W F

W W

F F F

F F

F

��

��

��

��

ii

ma ma

d d ma d K

K

d

d

外

外力重力

外力重力

重力力

重力外力

外力重力

(若移動過程中不 )加速

8.4: Determining Potential Energy values:

For the most general case, in which the force may vary with position, we may write the work W:

8.4: Determining Potential Energy values:

Elastic Potential Energy

In a block–spring system, the block is moving on the end of a spring of spring constant k. As the block moves from point xi to point xf , the spring force Fx =- kx does work on the block. The corresponding change in the elastic potential energy of the block–spring system is

If the reference configuration is when the spring is at its relaxed length,and the block is at xi = 0.

r(t) Rcos i + Rsin j ( t) 其中

( ) F (3 i + 4 j )N

( ) F (2y i + x j )N,

where x and y are in meters.

a

b

2

Find W for each one of the three paths.

(1) O A C(2) O B C(3) O C

8.5: Conservation of Mechanical Energy

Principle of conservation of energy:

In an isolated system where only conservative forces cause energy changes, thekinetic energy and potential energy can change, but their sum, the mechanical energy Emec of the system, cannot change.

The mechanical energy Emec of a system is the sum of its potential energy U and the kinetic energy K of the objects within it:

With and

We have:

8.7: Work done on a System by an External Force

Work is energy transferred to or from a system by means of an external force acting

on that system.


Work is energy transferred to or from a system by means of an external force acting on that system.

F F F

F F F

F F

F

F

F

W F

W F F F

��

��

��

��

��

��

��

��

ii

ma

d d d ma d K

d d d ma d K

K d d

ma

d

d

重力彈性力阻力



重力

外力

外

彈性力阻

力

力

力

外

外力

W F��

K U d 阻力外力


FRICTION INVOLVEDFRICTION NOT INVOLVED

thEKf kW f d

mec thE E Fd =

8.8: Conservation of Energy

Law of Conservation of Energy

The total energy E of a system can change only by amounts of energy that are transferred to or from the system.

where Emec is any change in the mechanical energy of the system, Eth is any change in the thermal energy of the system, and Eint is any change in any other type of internal energy of the system.

The total energy E of an isolated system cannot change.

mec; E + K U

Fig. P7-57, p.218

mJ

mT

(a) With what minimum speed must Jane begin her swing to just make it to the other side ?

(b) Once the rescue is complete, Tarzan and Jane must swing back across the river. With what minimum speed must they begin their swing ?

mecE +windW K U

Homework

• Chapter 7 ( page 161 )

25 , 27, 42, 48, 57, 65, 67, 75

• Chapter 8 ( page 190 )

15, 17, 19, 23, 27, 32, 33, 34, 40, 63, 78, 91

Due date 2014/ /

Chapter 7

is held in equilibrium in this final position

Chapter 8

Law of Conservation of EnergyExample

mec thE EW

The total energy E of an isolated system cannot change.

Eth is any change in the thermal energy of the system

Emec is any change in the mechanical energy of the system

There is a kinetic friction force, let = coefficient of kinetic friction

MVi

m

Frictionless floor

k

A

Consider a cart of mass M moves at a constant velocity Vi on a horizontal

frictionless floor. The block with a mass m dropped vertically on the cart at a point A. So the horizontal velocity of block “m” at the point A is zero. However, the kinetic friction force drives the block forward. This kinetic friction force causes the block to accelerate with horizontally.

ka gk kf mg

t=0 ( 時間 )

block

cart

MVi

m

Frictionless floor

t

Vf

v v

f i(M + m) M

Finally, at point “B” both the cart and block move with the same velocity . According to the law of momentum conservation at point B

v v

f i

M

M + m

v

f

AB

Let the time it takes the cart and block system to move from A to B be .

Using Newton’s 2nd law, acceleration of the block m is

MVi

m

Frictionless floor

t

kblock kma f mg

block ka g

vv 0 +i

f k

Mg t

M + m

v

( ) i

k

Mt

M + m g

Δt

Vf

• The change in kinetic energy of the whole system is

2 2

22

2

2

1 1( )v v

2 2

v1 1( ) v

2 2

1v 1

2

1v

2

f i

f i

ii

i

i

K K K

M m M

MM m M

M m

MM

M m

mM

M m

• Work done by the frictional force

?

?Kf

W l

l

k block ca

block

r

t

t

car

f

21v

2 ( )Kf i

mW M

M m

Let “d” be the displacement of the block ” m” relative to the floor while it is sliding on the cart.Using Work – Energy Theorem

k f if d k k

2 2v1 1v 0 [ ]

2 2i

k f

Mmgd m m

M m

2v1[ ]

2i

k

Md

g M m

block-floor

2 2

2 2

2 2

22

2

22

2

22

1 1Mv Mv

2 21 1

Mv Mv2 2

v1 1Mv [ ]

2 2

1Mv {1 }

2 ( )

1Mv {1 }

2 ( )

1 ( 2 )Mv

2 ( )

k f i

k i f

ik i

i

i

i

f D

f D

MmgD M

M m

M

M m

M

M m

m m M

M m

cart-

2

2floor

M( 2 )v

2 ( )i

k

m MD

g M m

Let “D” be the displacement of the cart ” M” relative to the floor (A→B) displacement while the block is sliding. Using Work – Energy Theorem

22

2

2

2

2

v Mv ( 2 )1( )

2 2 ( )

v1{ 2 }

2 ( )

v1

2 ( )

i iblock cart

k k

iblock cart

k

iblock cart

k

d D l

M m Ml

g M m g M m

MM m M l

g M m

Ml

g M m

block floor cart fl block caroor t

• Work done by the frictional force

2

2

v

2 ( )

1v

2 ( )

Kf k block cart

ik

k

i

W f l

Mmg

g M m

mM

M m

mec th th mecE EE E

• The change in kinetic energy of the whole system is

2 2

22

2

2

1 1( )v v

2 2

v1 1( ) v

2 2

1v 1

2

1v

2

f i

f i

ii

i

i

K K K

M m M

MM m M

M m

MM

M m

mM

M m

mec th th mecE EE E

Work done on a System by an External Force

Example

Law of Conservation of Energy

external force mec thE EW

MVi

m

Frictionless floor

Now we consider the same problem, but we will apply external force F to maintain the cart’s velocity (Vi). Since the kinetic friction force between the block and the cart is, net force on the cart must be zero to maintain the cart’s constant velocity, that is, . F k kf mg

There is a kinetic friction force, let = coefficient of kinetic friction

k

F

MVi

m

When block m becomes steady on the cart M, the force F disappears.

Frictionless floor

For the block “m”, the friction force acting on the block pushes it forward. So the block will slip on the cart. After a short time , slipping between the block and the cart ceased, and the block moves along with the cart, that is both move with the same velocity.

t

cart floorD

block cartL

F

• Using the equation of motion for the block with constant acceleration.

• During the period of time , the distance of the cart it moves relative to the floor is

0 0 kat g t v ik

tg

v i

( )k k

tg g

2v vv v i i

cart i iD

t

block k kma f mg

block ka g

• During this period of time , the distance of the block it moves relative to the floor is obtained from the following equation.

• During this period of time , work done by the external force F is

t

2

21 1 10 ( )

2 2 2block floor kk k

d t at gg g

2v vv i i

i

22( ) i

F cart floor k ik

vW FD mg mv

g

• Work done by the frictional force is

[ ]

1( )

2

1( )

2

1

2

Kf k

k

kk k

kk

W f L

f d D

mgg g

mgg

m

2 2i i

2

2

v v

v

v

block cart

block - floor cart - floor

i

i

kf

• The kinetic energy increase of the block is

1 10

2 2K m m 2 2

block v vi i

W W K F blockKf

Work done by the external force F

external force mec thE EW

0K cart

Conservative Forces and Potential Energy

• Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system

• The work done by such a force, F, is

U is negative when F and x are in the same direction

Slide 59

f

i

x

x f ixW F dx U U U


• The conservative force is related to the potential energy function through

• The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system– This can be extended to three dimensions

Slide 60

x

dUF

dx

Conservative Forces and Potential Energy – Check

• Look at the case of an object located some distance y above some reference point:

– This is the expression for the vertical component of the gravitational force

Slide 63

( )gy

dU dF mgy mg

dy dy

Energy Diagrams and Stable Equilibrium

• The x = 0 position is one of stable equilibrium

• Configurations of stable equilibrium correspond to those for which U(x) is a minimum

• x=xmax and x=-xmax are called the turning points

Slide 64

Energy Diagrams and Unstable Equilibrium

• Fx = 0 at x = 0, so the particle is in equilibrium

• For any other value of x, the particle moves away from the equilibrium position

• This is an example of unstable equilibrium

• Configurations of unstable equilibrium correspond to those for which U(x) is a maximum

Slide 65

• A stone of mass m = 8.0 kg is placed on the top of a relaxed spring, with spring constant , as shown in Figure. The stone is pushed down 40.0 cm and released. Choose the relaxed position of the top end of the spring as the origin (y = 0) and upward as the +y direction.

• Calculate the total energy of stone-spring-Earth system, taking both gravitational and elastic potential energies as zero for y = 0.

• Calculate the speed of the stone at y =0 after the release. • Determine the value of y ( measured from the origin y =0 ) for

which the kinetic energy of the stone is a maximum.

784 N/mk

8.8: Conservation of EnergyExternal Forces and Internal Energy Transfers

An external force can change the kinetic energy or potential energy of an object without doing work on the object—that is, without transferring energy to the object. Instead, the

force is responsible for transfers of energy from one type to another inside the object.

internal energy=biochemical in the muscles

2 20mv mv cos

K Fd

0

Force does no work

8.8: Conservation of EnergyExternal Forces and Internal Energy Transfers

An external force can change the kinetic energy or potential energy of an object without doing work on the object—that is, without transferring energy to the object. Instead, the

force is responsible for transfers of energy from one type to another inside the object.

internal energy=biochemical in the muscles

The net external force Fnet from the road change the kinetic energy of the car.

However, Fnet does not transfer energy from the road to the car and so does no

work on the car. Instead, the force is responsible for transfers of energy from the energy stored in the fuel.

Fnet=Σfriction ( f )


• Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system

• The work done by such a force, F, is

U is negative when F and x are in the same direction

Slide 74

f

i

x

x f ixW F dx U U U


• The conservative force is related to the potential energy function through

• The conservative force acting between parts of a system equals the negative of the derivative of the potential energy associated with that system– This can be extended to three dimensions

Slide 75

x

dUF

dx

Conservative Forces and Potential Energy – Check

• Look at the case of an object located some distance y above some reference point:

– This is the expression for the vertical component of the gravitational force

Slide 76

( )gy

dU dF mgy mg

dy dy

Energy Diagrams and Stable Equilibrium

• The x = 0 position is one of stable equilibrium

• Configurations of stable equilibrium correspond to those for which U(x) is a minimum

• x=xmax and x=-xmax are called the turning points

Slide 77

Energy Diagrams and Unstable Equilibrium

• Fx = 0 at x = 0, so the particle is in equilibrium

• For any other value of x, the particle moves away from the equilibrium position

• This is an example of unstable equilibrium

• Configurations of unstable equilibrium correspond to those for which U(x) is a maximum

Slide 78

• A stone of mass m = 8.0 kg is placed on the top of a relaxed spring, with spring constant , as shown in Figure. The stone is pushed down 40.0 cm and released. Choose the relaxed position of the top end of the spring as the origin (y = 0) and upward as the +y direction.

• Calculate the total energy of stone-spring-Earth system, taking both gravitational and elastic potential energies as zero for y = 0.

• Calculate the speed of the stone at y =0 after the release. • Determine the value of y ( measured from the origin y =0 ) for

which the kinetic energy of the stone is a maximum.

784 N/mk

Solution

• Stable equilibrium exists for a separation distance at which the potential energy of the system of two atoms (the molecule) is a minimum. Take the derivative of the function U(x):

Slide 80

12 6 12 6

13 7

( ) 12 64 4

dU x d

dx dx x x x x

Solution

• Minimize the function U(x) by setting its derivative equal to zero:

• Evaluate xeq the equilibrium separation of the two atoms in the molecule:

Slide 81

12 61/6

eq13 7eq eq

12 64 0 (2)x

x x

1/6 10eq (2) (0.263 nm) 2.95 10 mx

Solution• We graph the Lennard-Jones function on

both sides of this critical value to create our energy diagram as shown in Figure.

Slide 82

Solution

Slide 83

• Notice that U(x) is extremely large when the atoms are very close together, is a minimum when the atoms are at their critical separation, and then increases again as the atoms move apart. When U(x) is a minimum, the atoms are in stable equilibrium. indicating that the most likely separation between them occurs at this point.

Neutral Equilibrium

• Neutral equilibrium occurs in a configuration when U is constant over some region

• A small displacement from a position in this region will produce neither restoring nor disrupting forces

Slide 84

chapter 7 kinetic energy and work. 7.3 kinetic energy kinetic energy k is energy associated with the...

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