Chapter 7 Work and Kinetic Energy - University of Houstonnsmn1.uh.edu/rbellwied/classes/spring2013/ch7-notes.pdf · Title: ch7-notes.pdf Author: Rene Bellwied Created Date: 10/4/2011

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Chapter 7

Work and Kinetic Energy


Units of Chapter 7• Work Done by a Constant Force

• Kinetic Energy and the Work-EnergyTheorem

• Work Done by a Variable Force

• Power


7-1 Work Done by a Constant ForceThe definition of work, when the force isparallel to the displacement:

(7-1)

SI unit: newton-meter (N·m) = joule, J


7-1 Work Done by a Constant Force



If the force is at an angle to the displacement:

(7-3)


7-1 Work Done by a Constant ForceThe work can also be written as the dotproduct of the force and the displacement:



The work done may be positive, zero, ornegative, depending on the angle between theforce and the displacement:



If there is more than one force acting on anobject, we can find the work done by each force,and also the work done by the net force:

(7-5)


Example: A ball is tossed straight up. What is the work doneby the force of gravity on the ball as it rises?

FBD forrising ball:

x

y

w

Δr

ymg

ywWg

!"=

°!= 180cos


Example: A box of mass m is towed up a frictionless incline atconstant speed. The applied force F is parallel to the incline.What is the net work done on the box?

θ

F

w

NFx

y

θ

0cos

0sin

=!=

=!=

""

#

#

wNF

wFF

y

x

Apply Newton’s2nd Law:

An FBD forthe box:


The magnitude of F is: !sinmgF =

If the box travels along the ramp a distance of Δx thework by the force F is

!sin0cos xmgxFWF "=°"=

The work by gravity is

( ) !! sin90cos xmgxwWg "#=°+"=

Example continued:


Example continued:

The work by the normal force is:

090cos =°!= xNWN

The net work done on the box is:

0

0sinsin

net

=

+!"!=

++=

## xmgxmg

WWWW NgF


Example: What is the net work done on the box in theprevious example if the box is not pulled at constant speed?

!

!

sin

sin

wmaF

mawFFx

+=

="=#

Proceeding as before:

( )( ) xFxma

xmgxmgma

WWWW NgF

!=!=

+!"!+=

++=

net

net

0sinsin ##


7-2 Kinetic Energy and the Work-EnergyTheorem

When positive work is done on an object, itsspeed increases; when negative work is done,its speed decreases.



After algebraic manipulations of the equationsof motion, we find:

Therefore, we define the kinetic energy:

(7-6)


Example: The extinction of the dinosaurs and the majority ofspecies on Earth in the Cretaceous Period (65 Myr ago) isthought to have been caused by an asteroid striking the Earthnear the Yucatan Peninsula. The resulting ejecta causedwidespread global climate change.

If the mass of the asteroid was 1016 kg (diameter in therange of 4-9 miles) and had a speed of 30.0 km/sec,what was the asteroid’s kinetic energy?

( )( )

J 105.4

m/s 1030kg 102

1

2

1

24

23162

!=

!== mvK

This is equivalent to ~109 Megatons of TNT.



Work-Energy Theorem: The total work done onan object is equal to its change in kinetic energy.

(7-7)


7-3 Work Done by a Variable ForceIf the force is constant, we can interpret thework done graphically:


7-3 Work Done by a Variable Force

If the force takes on several successive constantvalues:



We can then approximate a continuously varyingforce by a succession of constant values.



The force needed to stretch a spring an amountx is F = kx.

Therefore, the workdone in stretchingthe spring is

(7-8)


Example: (a) If forces of 5.0 N applied to each end of aspring cause the spring to stretch 3.5 cm from its relaxedlength, how far does a force of 7.0 N cause the same springto stretch?

For springs F∝x. This allows us to write .

2

1

2

1

x

x

F

F=

Solving for x2: ( ) cm. 9.4cm 5.3N 5.0

N 0.7

1

1

2

2=!

"

#$%

&== x

F

Fx


Example continued:

N/cm. 43.1cm 3.5

N 0.5

1

1 ===x

Fk

(b) What is the spring constant of this spring?

N/cm. 43.1cm 4.9

N 0.7

2

2 ===x

Fk

Or


Example: An ideal spring has k = 20.0 N/m. What is theamount of work done (by an external agent) to stretch thespring 0.40 m from its relaxed length?

Fx (N)

x (m)x1=0.4 m

kx1

( )( ) ( )( ) J 6.1m 4.0N/m 0.202

1

2

1

2

1

curveunder Area

22

111====

=

kxxkx

W


7-4 Power

Power is a measure of the rate at which work isdone:

(7-10)

SI unit: J/s = watt, W

1 horsepower = 1 hp = 746 W


7-4 Power


7-4 Power

If an object is moving at a constant speed in theface of friction, gravity, air resistance, and soforth, the power exerted by the driving force canbe written:

(7-13)


Example: A race car with a mass of 500.0 kg completes aquarter-mile (402 m) race in a time of 4.2 s starting fromrest. The car’s final speed is 125 m/s. What is the engine’saverage power output? Neglect friction and air resistance.

watts103.92

1

5

2

av

!="

="

"=

"

"+"=

"

"=

t

mv

t

K

t

KU

t

EP

f


Summary of Chapter 7• If the force is constant and parallel to thedisplacement, work is force times distance

• If the force is not parallel to the displacement,

• The total work is the work done by the netforce:


Summary of Chapter 7

• SI unit of work: the joule, J

• Total work is equal to the change in kineticenergy:

where


Summary of Chapter 7

• Work done by a spring force:

• Power is the rate at which work is done:

• SI unit of power: the watt, W

Documents

Chapter 7 Work and Kinetic Energy - University of Houstonnsmn1.uh.edu/rbellwied/classes/spring2013/ch7-notes.pdf · Title: ch7-notes.pdf Author: Rene Bellwied Created Date: 10/4/2011