Work and Energy - Faculty Server...

Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11

Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI

Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html

Lecture 11

Chapter 7

Work and Energy(Work Done by a Constant Force)

10.16.2013Physics I


Chapter 7

Work done by a constant force Scalar (Dot) product of two vectors Kinetic Energy Work-Kinetic Energy theorem

Outline


Work Done by a Constant Force Work done on an object by a force If a constant force F moves object by a displacement d, work

done by force is given by W = F||d, where F|| is the component of the force along d

• Work has units of N-m, or Joules (J), and is a scalar !!

d


Example I

A force of 5 N, applied in the direction shown,moves a box across a horizontal distance of 10 m.How much work does the force do on the box?

NF 5

60

d 10m

= 25J


Scalar Product of Two Vectors

Three ways to multiply vectors

– Multiplication by a scalar

– Scalar (or dot) product

– Vector (or cross) product

vectorAc

scalarBA

vectorBA


Scalar (Dot) Product of Two Vectors

The scalar product of two vectors is written as:

Therefore, we can say that work is a scalar product of force and displacement

Work done by a force along a displacement:

Is it possible to do work on an

object that remains at rest?A) yes

B) no

Work requires that a force acts over a distance. If

an object does not move at all, there is no

displacement, and therefore no work done.

ConcepTest 1. To Work or Not to Work

The amount of work you actually do may have little relationship to the amount of effort you apply. For example, if you push on a car stuck in a snow drift, you may exert a lot of force (and effort) but if the car does not budge, you have not done any work! In order for work to be done on an object, the object must move some distance as a result of the force you apply.


Work

F

d

F

d

F

90d

W 0 W 0 W 0

f

N

mg

Displacement

Pull

Friction acts in the opposite

direction to the displacement, so

the work is negative. Or using the

definition of work (W = F d cos ),

because = 180º, then W < 0.

ConcepTest 2 Friction and Work I

A) friction does no work at all

B) friction does negative work

C) friction does positive work

A box is being pulled across a rough floor at a constant speed. What can you say about the work done by friction?

In a baseball game, the catcher

stops a 90-mph pitch. What can

you say about the work done by

the catcher on the ball?

A) catcher has done positive work

B) catcher has done negative work

C) catcher has done zero work

The force exerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F d cos ), because = 180º, then W < 0. Note that because the work done on the ball is negative, its speed decreases.

ConcepTest 3 Play Ball!

F

d

0W

Follow-up: What about the work done by the ball on the catcher?

ConcepTest 4 Tension and Work

A) tension does no work at all

B) tension does negative work

C) tension does positive work

A ball tied to a string is being whirled around in a circle. What can you say about the work done by tension?

v T

No work is done because the force

acts in a perpendicular direction to

the displacement. Or using the

definition of work (W = F d cos ),

because = 180º, then W < 0.

Follow-up: Does the Earth do work on the Moon?

ConcepTest 5 Force and Work

N

f

T

mg

Any force not perpendicularto the motion will do work:

N does no work

T does positive work

f does negative work

mg does negative work

1) one force

2) two forces

3) three forces

4) four forces

5) no forces are doing work

A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?


Work on a backpack. Example 7-2

(a) Determine the work a hiker must do on a 15.0-kg backpack to carry it up a hill of height h = 10.0 m, as shown. Determine also

(b) the work done by gravity on the backpack, and

(c) the net work done on the backpack. For simplicity, assume the motion is smooth and at constant velocity (i.e., acceleration is zero).


Work on a backpack.

FH d

mg

θ

180‐θ


Work on a backpack.

FH d

mg

It is not an accident that the net work is zero.

Wherever, the net force acting on an object is zero, the net work will be zero as well

321 WWWWnet

dFFFdFdFdF

321321 dFnet

0,0 netnet WthenFIf


Scalar Product (component form)

kAjAiAA zyxˆˆˆ

kBjBiBB zyxˆˆˆ

In component form:

zzyyxx BABABABA

A

B


Scalar Product Math kAjAiAA zyxˆˆˆ

kBjBiBB zyxˆˆˆ

A

B (Axi Ay j Azk)(Bxi By j Bzk) AxBx (i i ) AxBy (i j) AxBz (i k)

AyBx ( j i ) AyBy ( j j) AyBz ( j k)

AzBx (k i ) AzBy (k j) AzBz (k k)

(i i ) ( j j) (k k) 1

)ˆˆ( ji

Proof:

(i j) (i k) ( j k) 0

90cosˆˆ ji 00

The scalar product of two unit vectors)ˆˆ( ii 0cosˆˆ ii 1

11

Similar andUsing these we can write:

0 0

0 0

0 0

1

1

1

zzyyxx BABABABA


Properties of Scalar Product

• Commutative property ABBA

CABACBA

• Distributive property


Example

A constant force F acts on an object as it moves from position x1 to x2. What is the work done by this Force?

kjix ˆ3ˆ4ˆ21 kjix ˆ6ˆ3ˆ52

kjiF ˆ2ˆ5ˆ4

xd

JdFW 29)9)(2()7)(5()3)(4(

12 xx kji ˆ9ˆ7ˆ3


Translational Kinetic Energy

Kinetic energy (K) is the energy of motion Translational kinetic energy is the energy of motion in a

line or trajectory Rotational kinetic energy (Ch. 10) is the energy of

circular/rotational motion

Kinetic energy is the energy possessed by an object because

of its motion


Kinetic Energy I

A constant horizontal force F acts on an object on a frictionless surface. The box accelerates, i.e. speeds up.

We want to define a property of the object (called kinetic energy, or energy of motion) which changes as work is done on it by the force.

F Fd

vi vf


Kinetic Energy II

F Fd

vi

F mavf

2 vi2 2ad

dmFvv if

222

K f Ki W

12

mvf2 1

2mvi

2 Fd

12

mvf2 1

2mvi

2 Fd

mvf2 mvi

2 2Fd K 12

mv2

K f Ki K Wnet

Kinetic energy

Work-Kinetic Energy Principle

The work done is equal to the change in the kinetic energy.

vf


Work-Energy Principle

K f Ki K Wnet


If the net work is positive, the kinetic energy increases.

If the net work is negative, the kinetic energy decreases.

Work and kinetic energy have the same units: joules.

Energy can be considered as the ability to do work.

A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child?

1) positive work was done 2) negative work was done 3) zero work was done

The kinetic energy of the child increased because her speed increased. This increase in KE was the result of positive work being done.

ConcepTest 6 Work and KE

Follow-up: What does it mean for negative work to be done on the child?


ExampleNet Work required to accelerate a 1000kg car from

KW

kJJvvmKW if 800000,800]400[2

1000][2

222

(a) from 20 m/s to 40 m/s?

(b) from 40 m/s to zero?

vi vf

][2

22if vvm

kJJ 600000,600]2040[2

1000 22

K f Ki K Wnet



Work & Kinetic Energy

The work done by the earth on an apple as the apple falls towards the earth with increasing speed v is positive.So kinetic energy increases.

The work done by the earth on the moon as the moon makes one full revolution around the earth at constant speed v is zero.So there is no change in kinetic energy.


Thank youSee you on Monday

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