ch.11 energy i: work and kinetic energy ch. 11 energy i: work and kinetic energy

Ch.11 Energy I:Work and kinetic energy

Ch. 11 Energy I:Work and kinetic energy


11-1 Work and energy

Example: If a person pulls an object uphill. After some time, he becomes tired and stops.

We can analyze the forces exerted in this problem based on Newton’s Laws, but those laws can not explain: why the man’s ability to exert a force to move forward becomes used up.

For this analysis, we must introduce the new

concepts of “Work and Energy”.


Notes: 1) The “physics concept of work” is different from the “work in daily life”;

2) The “energy” of a system is a measure of its capacity to do work.


11-2 Work done by a constant force

1.Definition of ‘Work’

The work W done by a constant force that moves a body through a displacement in the directions of the force as the product of the magnitudes of the force and the displacement:

(11-1)

F

s

(Here )FsW F//s

11-3 Power


The normal force does zero

work; the friction force does

negative work, the gravitational

force does positive work which is

or

N

N

mghmgs cos

)cos(cos mgsmgs

gm

gm

f

shv

Fig 11-5

f

Example: In Fig11-5, a block is sliding down a plane.


2. Work as a dot product

The work done by a force can be written as

(11-2)

(1) If , the work done by the is zero.

(2) Unlike mass and volume, work is not an intrinsic

property of a body. It is related to the external force.

(3) Unit of work: Newton-meter (Joule)

(4) The value of the work depends on the

inertial reference frame of the observer.

F

W F s

sF

F

gm

shv


If a certain force performs work on a body in a time , the average power due to the force is

(11-7)

The instantaneous power P is

(11-8)

If the power is constant in time, then .

av

WP

t

dWP

dt

avPP

3. Definition of power: The rate at which work is done.

Wt


(11-10)

Unit of power: joule/second (Watt)

dW F d s d sP F F v

dt dt dt

If the body moves a displacement in a time dt,

sd

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Work done by a variable force

1.One-dimensional

situation

The smooth curve in

Fig 11-12 shows an

arbitrary force F(x) that

acts on a body that

moves from to .

Fig 11-12

ix

ixfx

fxx

x

F

1F

2F )(xFx

11-511-4


We divide the total displacement into a number N

of small intervals of equal width . This interval so

small that the F(x) is approximately constant. Then

in the interval to +dx , the work

and similar ……The total work is

or

(11-12)

1 1W F x

x

2 2W F x 1x1x

N

nn xFW

1

...... 2121 xFxFWWW


To make a better approximation, we let go to zero and the number of intervals N go to infinity. Hence the exact result is

or

x

01

limN

nx

n

W F x

01

lim ( )f

i

N x

n xxn

W F x F x dx

(11-13)

(11-14)

ix fx

Numerically, this quantity is exactly equal to the area between the force curve and the x axis between limits and .


Example: Work done by the spring force

In Fig 11-13, the spring is in the relaxed state, that is no force applied, and the body is located at x =0. o x

Relaxed length

Fig 11-13

2 21( )

2

f

i

x

s s f ixW F dx kxdx k x x

kxFs Only depend on initial and final positions


Fig 11-16 shows a particle moves along a curve from to f . The element of work

The total work done is (11-19)

i

f

F

sd

x

y

o

i

dW F d s

cosf f

i iW F d s F ds

( )( )

( )

f

x yi

f

x yi

W F i F j dxi dy j

F dx F dy

Fig 11-16

2.Two-dimensional situation

(11-20)

or


Sample problem 11-5

A small object of mass m is

suspended from a string of length L.

The object is pulled sideways by a

force that is always horizontal, until

the string finally makes an angle

with the vertical. The displacement is

accomplished at a small constant

speed. Find the work done by all the

forces that act on the object.

y

mgm

F

m

dsT

Fig 11-17

F

mx


11-6 Kinetic energy and work-energy theorem

for a body of mass m moving with speed v.

21

2K mv

,F

,a

v Relationship between

Work and Energy1. Definition of kinetic energy K:

2. The work-energy theorem:

2 21 1

2 2netF f iW mv mv (11-24)

“The net work done by the forces acting on a body is equal to the change in the kinetic energy of the body.”


3. General proof of the work-energy theorem

For 1 D case: represents the net force acting on the body.

netF

dx

dvmv

dt

dx

dx

dvm

dt

dvmmaF x

xxx

xnet

The work done by isnetF

xnet net x

dvW F dx mv dx

dx

It is also true for the case in two or three dimensional cases

2 21 1

2 2

xf

xi

v

x x xf xi

v

mv dv mv mv


4.Notes of work-energy theorem:

2 21 1

2 2netF f iW mv mv

The work-energy theorem survives in different inertial reference frames.

But the values of the work and kinetic energy in their respective reference frames may be different.

Please relate a) point to conservation of momentum

a). In different inertial reference frames?

b). Limitation of the theorem

It applies only to single mass points.


11-7 Work and kinetic energy in rotational motion

1.Work in rotation

Fig11-19 shows an

arbitrary rigid body to

which an external agent

applies a force at

point p, a distance r

from the rotational axis.

Fig 11-19

x

y

Pr

F

F

d

O

ds


As the body rotates through a small angle

about the axis, point p moves through a

distance . The component of the force in

the direction of motion of p is ,and so the

work dw done by the force is

rdds

sin

( sin )( )

( sin )

dW F ds

F rd

rF d

d

sinF

zd

Fr


So for a rotation from angle to angle, the work in the rotation is

(11-25)

The instantaneous power expended in rotation

motion is

i f

f

izW d

z z z

dW dP w

dt dt

(11-27)


2. Rotational kinetic energy

Fig 11-20 shows a rigid body rotating about a fixed axis with angular speed

. We can consider the body as a collection of N particles , ……

moving with tangential speed , …… If indicates the distance of particle from the axis, then and its kinetic energy is

. The total kinetic energy of the entire rotating body is

1m 2m

2v1v

nm nn rv

222

2

1

2

1 nnnn rmvm x

y

O

nr

1m

2m

2r1r

ω

Fig 11-20


is the rotational inertia of the body,

then

22

2222

2211

)(2

12

1

2

1

nnrm

rmrmK

Irm nn 2

2

2

1 IK

(11-28)

(11-29)

3. The rotational form of the work-energy theorem

W K

which can be obtained similarly as for single particles.


Table 11-1

Translational quantity

Rotational quantity

Work

Power

Kinetic energy

Work-energy theorem W K

2

2

1 IK

W K

zzP

2

2

1mvK

xxvFP

xW F dx zW d


Sample problem 11-10

A space probe coasting ( 航线 ) in a region of negligible gravity is rotating with an angular speed of 2.4rev/s about an axis that points in its direction of motion. The spacecraft is in the form of a thin spherical shell of radius 1.7m and mass 245kg. It is necessary to reduce the rotational speed to 1.8rev/s by firing tangential thrusters ( 推进器 ) along the equator of the probe.

What constant force must the thruster exert if the change of angular speed is to be accomplished as the probe rotates through 3.0 revolution?


F

v

Solution: For a thin spherical shell

The change in rotational kinetic energy is

222 472)7.1()245(3

2

3

2mkgmkgMRI

2 2

4

1 1

2 2

2.67 10

f iK I I

J

Z

Nrevm

J

R

KF 833

)]0.3()2[()7.1(

1067.2 7

zW RF K

in –z directionz

The rotational work is

then


11-8 Kinetic energy in collision

We reconsider a collision between two bodies that move

along the x axis with the analysis of kinetic energy.

1. Elastic collision: the total kinetic energy before

collision equals the total kinetic energy after the

collision.

fi kk (11-30)


Fig (6-17) Two-body collisions in cm frame

1minitial

elastic

inelastic

Completely inelastic

explosive

1.

3.4.5.

2m'

1iP'

2iP

Final

2.

'1fP

'

2fP


2. Inelastic collision: the total final kinetic energy is less

than the total initial kinetic energy. (If you drop a tennis ball on a hard surface, it does not quite bounce to its original height.)3. Completely inelastic collision: two bodies stick together. This type of collision loses the

maximum amount of kinetic energy, consistent with the conservation of momentum.

4. Explosive or energy releasing collision: “The total final kinetic energy is greater than the total initial kinetic energy.” Often occur in nuclear reactions.

ch.11 energy i: work and kinetic energy ch. 11 energy i: work and kinetic energy

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