6 Work and Kinetic Energy

• Work Done by a Constant Force

• Work Done by a Variable Force – Straight Line Motion

• The Scalar Product

• Work-Kinetic Energy Theorem – Curved Paths

• Hk: 27, 41, 49, 51.

Energy: The work that a physical system is capable of doing in changing from its actual state to a specified reference state … (American Heritage Dictionary)

Energy: The capacity to do work. (Physics)

What is Work?

Some Definitions

Work Transformation

• Work is a usage of energy, e.g.,

• Burning gasoline produces heat & motion

• battery running a car

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Work

• Work is force x distance (N·m = joule), force parallel to motion (no work done by perpendicular component)

• It takes energy to do work.

• Less stored energy is available after productive work is done.

Work by Constant Force

xFxFW x cos

Work = Fcosx = (80N)(cos40)(11m) = 674 J

Ex: F = 80N, Angle is 40°, x is 11m,

Total Work on Object

xFxFxFW xnetxxtotal ,21

xxnet maF , x

vva ifx

2

22

xmaW xtotal

2212

21

22

2 ifif

total mvmvxx

vvmW

Kinetic Energy

Energy Kinetic of Definition

221 mvKE

TheoremEnergy -Kinetic -Work

KEWtotal

Example: 20kg moving at 5m/s. 250J of work (total) are done on it. What is its final speed?

2212

21

iftotal mvmvW 2

212

21 5)20()20(250 fv

251010250 2 fv25010250 2 fv

210500 fv 502 fv

Negative Work (object slows down)

negative becan cos cos xFW

Ex. A car moves 10 meters while a braking force of 500 newtons acts.

Energy Kinetic of 500Jlost car

50010)180(cos500 JmNW

Ex. Block pushed 3m with 75N of force while Friction of 50N. Total Work is,

JmN

mNN

xFW xnettotal

75)3)(25(

)3)(5075(,

Work by a Variable Force, Straight Line Motion

with xchanges force where

2

1x

x xdxFW

1

0 212

212

21

1

0

221 01

xalong 1 to0 moving Ex.

|xxdxW

xFx

Hooke’s Law

• Elastic restoring force proportional to deformation

• F = -kx k = elastic constant (N/m)

• Ex. Lab springs, k = 8N/m, 0.1kg mass:

• mg = kx

• (0.1kg)(9.8N/kg) = 8N/m(x)

• x = 0.98N/(8N/m) = 0.1225 m

Scalar (Dot) Product

zzyyxx BABABABA

ProductScalar - Definition

cosABBA

)ˆˆˆ()ˆˆˆ( kBjBiBkAjAiABA zyxzyx

WorklIncrementa of Definition

)(cos

dFdFdW

5)0)(0()0)(1()5)(1( BA

Ex: A = (1, 1, 0), B = (5, 0, 0)

552

45cos52

cos

22

ABBA

2011 222 A5005 222 B

45

5)0)(1()0)(1()5)(1( BA

cosABBA 3111 222222 zyx AAAA

5005 222222 zyx BBBB

cos535

7.54

3

1

53

5cos

Example: Find the angle between A = (1, 1, 1) and B = (5, 0, 0)

FFsFW 4)0,3,4)(0,0,(

Power

J/s tt Power wa of Definition

dt

dWP

vFdt

dF

dt

dWP

watt746 lb/sft 550 hp 1

Ex: A car drives at 20m/s and experiences air-drag of 400N.

wattsmNvFP 8000)/20)(400(

hpwatt

hpwatt10

746

1

1

8000

What size motor needed when Operating Speed is 10cm/s?

Cube of bricks ~ 1 ton

1 ton = 2000 lbs ~ 9000 N

Minimum Power:

P = Fv = (9000N)(0.1m/s)

P = 900 W = 1.2 hp

Work along Curved Path

2

1

2

1dtvFdFW netnettotal

2

1

2

1

2

1vdvmdtv

dt

vdmdtvam

12

2

1

221 | KEKEmvWtotal

Summary

• Work is force parallel to path x distance (force constant)

• Negative total work (object slows down)

• Work is integral of force·distance (Scalar Product)

• Power is rate work is done

• Total work = change in KE

• /