1

New policy for workshops

You are expected to print, read, and think about the workshop material prior to coming to class. (This part of the policy is not new!) There will be a “prelab” question which is to be answered on the worksheet before coming to class. The prelab question will be graded during the quiz. (This policy replaces the penalty for not printing the worksheet in advance.)

Work & Kinetic Energy

Review of Previous Lecture

• Fluid resistance, terminal speed

at high speed

• Uniform circular motion:

• If net force on object is perpendicular to v:

• Net force causes radial acceleration

via Newton’s 2nd law.

• Object’s path becomes circular with

radius R:

Tangent to circle ; v = const

Radially inward ; a = const

v

Dmgvmgf

Dvf

/

2

v

a

var

2

Rv

ra

RmvF

2

a

vR

rF

v

Rm

• The net force pointing to the center is called “radial” or

“centripetal” force.

• The centripetal force may be made up of weight,

friction, tension, normal forces acting on object – it is

not an additional force!

A block is given a kick and then freely slides up an inclined plane at angle q with the horizontal. There is friction between the block and plane. The magnitude of the acceleration of the block is

A. g

B. g sinq

C. g cosq

D. larger than g sinq

E. smaller than g sinq

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A B C

A soccer ball is dropped from a hot air balloon. During the fall, air resistance cannot be neglected. Which of the following graphs best represents the acceleration vs. time graph? (Choose down direction as positive.)

A ball is attached to the end of a cord of length 2 meters. The ball is whirled in a vertical circle. Which is correct?

A. The tension at the bottom and at the top is greater than the weight of the ball.

B.The tension at the bottom and the top is less than the weight of the ball.

C.The tension at the bottom is greater, and at the top less than the weight.

D. The tension at the bottom is greater than the weight, and at the top it may be greater or less than the weight.

E. The tension at either location may be greater or less than the weight, depending on the speed.

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A 2 kg ball is attached to the end of a cord of length 2 meters. The ball is whirled in a vertical circle. The tension in the cord at the top of the circle is 0 N. Which is correct?

A. The acceleration at the top of the circle is zero.

B. The acceleration at the top of the circle is 9.8 m/s2, upward.

C. The acceleration at the top of the circle is 9.8 m/s2 downward.

D.The acceleration at the top of the circle has magnitude 19.6 m/s2.

E. The acceleration at the top of the circle cannot be determined.

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VECTOR MULTIPLICATION:

SCALAR PRODUCT

Vector Addition:

What does mean?

Scalar Product (= Scalar)

(projection of on ):

Note:

Scalar product of two vectors is COMMUTATIVE.

BA

BA

BA

BA

qcosABBA

ABA

B

A

BAAB

qq

q

coscos

cos

ABBA

ABAB

A

B

A B

A

B

BA

BA

A

B

qcosB

q

qcosA

qB

A

SCALAR PRODUCT USING COMPONENTS

Dot product is distributive

Consider:

Use dot product to find angle between vectors.

Example:

)( CBA

CABA

jAiAA yxˆˆ

jBiBB yxˆˆ

)ˆˆ()ˆˆ( jBiBjAiABA yxyx

jjBAijBAjiBAiiBA yyxyyxxxˆˆ ˆˆ ˆˆ ˆˆ

)( zzyyxx BABABA

j

1ˆˆˆˆ jjii

0ˆˆ ji

2|| ; ˆˆ AkiA

2|| ; ˆˆ BkjB

1)ˆ()ˆˆ( kjkiBA

cos|||| BABA

!60 cos2

1

||||cos

211 o

BA

BA

A

B

BA

CA

C

CB

A

B

1 0 0 1

i

J )3)(cos5(

ˆ)m3(]ˆ)sinN5(ˆ)cosN5[(

ˆ)inN5(ˆ)cosN5(

q

qq

qq

ijiW

jsiF

WORK

Suppose block of mass m lies on horizontal plane.

Case A: A horizontal

force acts continuously

as block is displaced by

Work done by force:

Case B: Force and

displacement are

not collinear

Work = (displacement) X (component of F along displacement)

EXAMPLE:

F

s

WsFsFW

)cos( q

J 15 Nm 15]ˆ)m3[(]ˆ)N5[(

ˆ)m3( , ˆ)N5(Let

iiW

isiF

F

m

s

F

m

s

q

A:

B:

FsW

Work done (by a force) may be:

• Positive

• Negative

• Zero

EXAMPLES:

• Positive: e.g. block on previous viewgraph

• Negative:

Arnold lowers

the barbell

• Zero: Uniform circular motion

tangent ||

radial

vsd

Fc

F

s

v

CF

n – mgcosq = 0

n = mgcosq

mgsinq – fK

fK = mKn = mKmgcosq

qmqm cosˆˆcos mgsisimgsfW KKKf

jmgimgwFGravˆcosˆsin qq

qsinmgsswWg

EXAMPLE:

A block of mass m slides

a distance s down an incline

plane.

How much work is done

by friction?

How much work is done by gravity?

i

yi

F

i

xi

F

s

q

m

m

n

q mgcosq

mgsinq

fK

mg

What is the net work done by all forces acting on the

block as it slides?

jmgnimg

jFiFFF

K

i i i

yxinet ii

ˆ)cos(ˆ)cos(sin

ˆ)(ˆ)(

qqmq

qmq cossin Kx mgssF SFW netnet

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Constant speed!!!

Net force = 0

Work = 0

n

mg

T

Fk

s

A block is pushed so that it moves distance L up a ramp (incline angle q ) at constant speed. If there is friction, the work done on the block by the hand…

A. is mg sinq L.

B. is less than mg sinq L.

C. is greater than mg sinq L.

D. could be greater of less.

E. is zero.

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Is the work positive or negative?

A block is pushed so that it moves distance L up a ramp (incline angle q ) at constant speed. The work done on the block by the normal force of the ramp…

A. is mg cosq L.

B. is less than mg cosq L (but > 0)

C. is greater than mg cosq L.

D. could be greater of less.

E. is zero.

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A block is pushed so that it moves distance L up a ramp (incline angle q ) at constant speed. If there is friction, the magnitude of the work done on the block by friction…

A. is mg sinq L.

B. is less than mg sinq L.

C. is greater than mg sinq L.

D. could be greater of less than mg

sinq L.

E. is zero.

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Is the work positive or negative?

A block is pushed so that it moves distance L up a ramp (incline angle q ) at constant speed. If there is friction, the magnitude of the work done on the block by gravity…

A. is mg sinq L.

B. is less than mg sinq L.

C. is greater than mg sinq L.

D. could be greater of less than mg

sinq L.

E. is zero.

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Is the work positive or negative?

A block is pushed so that it moves distance L up a ramp (incline angle q ) at constant speed. If there is an unknown amount of friction, the net work done on the block…

A. is zero.

B. is less than zero.

C. is greater than zero.

D. could be positive or negative.

E. need to know the friction force.

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2

1212

221 mvmvWtot

2

21 mvK

Work done by a net force on an object of mass m = change in kinetic energy of the object

WORK AND KINETIC ENERGY

Consider collinear force and displacement:

Where Fnet is NET force

RECALL: If constant acceleration a

changes v1 to v2 over a distance s

Define Kinetic Energy:

Work-Energy Theorem:

sFW nettot

a

vvs

2

2

1

2

2

2

2

1

2

2 vvmsma

Fnet

2

1212

221 mvmvKsFW nettot

m 30

m/s 30kg 10002

21

netF

m

/sm kg

30

900500 22

2

5

s

m kg

30

105.4

N 105.1 4

24

m/s 15kg

N

1000

105.1

a

m

Fnet

What is the corresponding acceleration???

EXAMPLE:

Recall the yellow light

Car with Mass of

1,000 Kg moving at

30 m/s must stop in 30 m.

How much force must be applied?

jsjmgFnetˆ)m 320( ;)ˆ(

2

1212

221)m 320( mvmvmgWtot

)m 320(22

2 gv

/sm 6400 22

m/s 802 v

EXAMPLE:

Drop a bowling ball from the top

of the Empire State Building.

How fast is it going just before it

hits the ground?

Use Work-Energy Theorem:

KSFW nettot

m 320

Two identical blocks are released from rest at the same

height on two separate frictionless ramps. The blocks

reach the same final height labeled finish. Which

statement is correct regarding the work done on the

two blocks by the gravitational force?

A. Block B travels further, so more work is done on block B by the gravitational force than on block A.

B. Both blocks fall the same vertical distance, so the work done by gravity is the same.

C. By Newton’s Third Law, the force exerted on the block by the earth is exactly cancelled by the force exerted on the earth by the block. The work done is zero.

D. The angle between force and displacement is smaller for block A than for block B, thus the work done on block A is greater.

E. None of the above.

i-Clicker

Two identical blocks are released from rest at the same

height at the same time. Block A slides down a steeper

ramp than Block B. Both ramps are frictionless.

The speed of Block A as it crosses the finish line is:

A. greater than the speed of Block B at the finish line.

B. less than the speed of Block B at the finish line.

C. equal to the speed of Block B at the finish line.

D. Not enough information.

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Constant force K = Fs

Same constant force acts over same distance

K1 = Fs = K2

i

iix

tot xxFW )(lim0

2

1

)(x

xtot dxxFW

WORK DONE BY VARYING FORCE

• Consider varying force along line

For constant force:

For varying force:

EXAMPLE: Force to stretch spring

W = Fs

W = Area under

Force vs. displacement

curve

k = spring constant

[N/m]

“Hooke’s Law”

kxFx

Fx

x1 x2

x s

x 2x

F

2F

Fx

x

F(x)

x1 x2

xi

F(xi)

F

2

1

2

1

2

21

x

x

x

xkxdxkx

)( 2

1

2

221 xxk

Work done by F to stretch spring:

In General: Work done on spring to extend from x1 to x2:

NOTE: Work-Energy theorem valid for varying

forces on curved paths!

2

1

)(x

xx dxxFW

2

21

0

2

21 kXkxW

X

X

o

X

ox

kxdx

dxxFW

)(Fx

x

F(x)

X

kX

10 cm

)( 2

1

2

221 xxk

Wtot = 4.00 J

k = 800 N/m

Initial extension:

Additional extension:

22

21 )cm 10()cm 20( kW

22

21 m 010.0m 040.0)N/m 800(

= 12.0 J

EXAMPLE: It takes 4.00J to stretch a spring from

equilibrium to 10.0 cm. How much work must be

done to stretch it an additional 10 cm?

22

21 )m 0.0()m 10.0( k

27

New policy for workshops

You are expected to print, read, and think about the workshop material prior to coming to class. (This part of the policy is not new!) There will be a “prelab” question which is to be answered on the worksheet before coming to class. The prelab question will be graded during the quiz. (This policy replaces the penalty for not printing the worksheet in advance.)