engr221 lecture 27

8/7/2019 ENGR221 Lecture 27

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Kinetics of Particles Kinetics of ParticlesENGR 221

April 23, 2003


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Lecture Goals

Lecture Goals

Energy Wo rk Examples


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W ork and Energy Methods

W ork and Energy Methods

Wo rk /Energy meth od is an alternative meth od of

so

lving the equatio

nsof

mo

tio

nof

particles.It co mbines the kinematics with Newt o ns 2nd L aw,

by integrating Newt o ns law with respect t o po sitio n.(fo rce as a f uncti o n of po siti o n).


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Energy Method Energy Method Given a bl o ck with a fo rce P

actingo

n it

x x x

y y

co s sinco s sin 0

F ma P mg R ma F ma N mg P

U UU U

! p !! p !

P


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Definition of Acceleration

Definition of Acceleration

T he accelerati o n is d ef ine d as

1vs

0 0

dv dv dxa

dt dx dt

adx vdv

adx vdv

! !

!!


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Kinetic Energy Kinetic EnergyT he right han d sid e of the equati o n is kn o wn as the

Kinetic Energy

T he units in S I are J o ules (N-m) o r (English) f t-lbs

212

T m v!


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Kinetic Energy Kinetic EnergyT he energy is path d epen d ent s o that the tangentialaccelerati o n can be use d to f ind

t tt

t t t

dv dv dsa

dt ds dt a ds v dv

! !

!


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Kinetic Energy Kinetic EnergyT he le f t sid e of the equati o n of fo rces can be m od if ied to use an d the resultant of the integrati o ns are scalar values.

2

1

vs

0 v

2 22 1

co s sin

2 2

P mg R d s mvd v

v vm

U U !

!


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W ork

W ork

Wo rk do ne (U) by the fo rce is the pr od uct of a magnitu d eof P and the rectangular c o mp o nents of the d isplacementin d irecti o n of the fo rce d co s J .

x y z

- S calar Quantities

co s

U P d

d U P dx P ds

P dx P dy P dz

J

! y

! y!

!

ur ur

ur uur


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W ork

W ork

T he w o rk do ne by the system is d ef ine d as.

2 2 2

1 1 1

2

1 21

x y z

x y zx y z

U d U

P d x P d y P d z

p !

!


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W ork

W ork

If the c o nstant fo rce P

2 2 2

1 1 1

2 2 2

1 1 1

x y z

1 2 x y zx y z

x y z

x y zx y z

x 2 1 y 2 1 z 2 1

U P dx P dy P dz

P dx P dy P dz

P x x P y y P z z

p!

!

!


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Principle of W ork and Kinetic Principle of W ork and Kinetic Energy Energy

T he principle of wo rk an d kinetic energy is d ef ine d as aco nservative system.

{ { {1 1 2 2initial kinetic f inal kineticwo rk

energy energydo ne

T U T p

!


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Problem Solving Technique Problem Solving TechniqueT o so lve a particle d ynamics pr o blem with w o rk-energymeth od

Draw the f ree-b od y d iagram of the o bject an d inclu d e all fo rces acting o n the o bject.

List the

forces an

dcalculate the w

ork

done by eachfo rce.

Determine the initial an d f inal kinetic energy.


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Example Problem Example Problem Energy Energy

A 45 kg bl o ck is sli d e a d istance of 3 malo ng a h o rizo ntal sur f ace by a c o nstantfo rce P = 200 N , incline 30 o to ho rizo ntal. T he c o eff icient of slid ingf ricti o n is Q = 0.30

Determine the n o rmal an d tangential fo rces the sur f ace exertso n the bl o ck.

Determine the wo

rk do

ne by eachfo

rce that actso

n the blo

ck as it sli d es 3 m.

Determine the t o tal w o rk do ne o n the bl o ck.

Assuming that the bl o ck starts f r o m rest, d etermine the spee d of the bl o ck when it has travele d 3 m.

1)

2)

3)

4)


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Example Problem Example Problem Energy EnergyDraw the f ree-b od y d iagram of the pr o blem, the bl o ck will sli d eo nly in the x d irecti o n there fo re

o

y

2 osin 30 0

45 kg 9.81 m/s 200 N sin 30

341.45 N

F N W P

N

N

! ! !

!


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Example Problem Example Problem Energy EnergyT he fo rce F is d ef ine d as

0.30 341.45 N102.44 N

F N Q! !

!


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T he w o rk do ne by the c o mp o nents

w

N

o o

P

441.45 N 3 m i 0

341.45 N 3 m i 0

102.44 N i 3 m i 307.32 N -m

200 N co s 30 i 200 N sin 30 3 m i 519.62 N -m

U F s

U

U

U

U

! y

! y !

! y !! y !

! y !

r r

r r

r r

r r

r r r


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T he kinetic energy(initially) is zer o ,

so

1 1 2 2

2

2

2

0 212.3 N

-m 22.5 kg3.07 m/s

T U T

vv

p !

! !


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T he kinetic energy(initially) is zer o ,

so

T his is the same as

1 1 2 2T U T

p !

2 22 2 222

1 145 kg

2 222.5 kg

T mv v

v

! !

!


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C lass Problem

C lass Problem Energy Energy

Draw the f ree-b od y d iagram of the pr o blem


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Example Problem Example Problem W

ork W

ork A bushing sli d es o n a sha f t thatfo rms a 30 o angle with respect t o

the x axis. It is acted

o

n by aco nstant fo rce

Determine the w o rk the fo rce per fo rms o n the bushing in the timethe bushing sli d es 3 m al o ng thesha f t.

2 3 N F i j! ur r r


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ork W

ork T o co mpute the U 1--> 2 Use theintegral of the path. S o

o oco s 30 sin 303 1

2 2

i j

i j

P !

!

ur r r

r r


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ork W

ork T o co mpute the U 1--> 2 Use the integral of the path.S o

2 3 m

1 0

3 m

0

3 m

0

3 12 3 N

2 23 1

2 32 2

d U F d

i j i j d s

d s

P! y

! y

!

ur uuur

r r r r


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ork W

ork T o co mpute the U 1--> 2 Use the integral of the path.S o

? A

? A

2 3 m 3 m

1 0 0

3 12 3 3.232 N

2 2

3.232 N 3.0 m 9.70 N-m9.70 J o ules

d U ds ds ! !

! !!


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Example Problem Example Problem Energy EnergyT he w o rk o n do ne by the tensi o n T isthe negative of the w o rk do ne by that

of blo ck B, the net w o rk do ne by T iszer o . T he w o rk of the system is

1

2

T

T

T i 5 f t i 5 T f t-lb

T j 5 f t j 5 T f t-lb

U

U

! y !

! y !

r r

r r


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Example Problem Example Problem Energy EnergyT he o ther c o mp o nents of wo rk are

T he f inal w o rk do ne is

2 o

6 slug

F

24 slug

6 slug 32.2 ft/s sin 30 i 5 ft i 483 ft-lb16.73 lb i 5 ft i 83.65 ft-lb

4 slug 32.2 ft/s 5 ft 644 ft-lb

U

U

U

! y

! ! y !

! y !

r r

r r

r r

1 2 483 f t-lb 83.65 f t-lb 644 f t-lb

77.35f t-lb

U p

! !


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Example Problem Example Problem Energy EnergyT he initial kinetic energy is zer o and the f inal

1 1 2 2

2

2

10 77.35 f t-lb

21 6 slug 4 slug 77.35 f t-lb2

3.93 f t/s

T U T

m v

v

v

p !

!

!

!


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C lass Problem


T wo blo cks are j o ine d by aninextensible cable. I f the systemis release d f r o m rest, d eterminethe vel o city of blo ck A a f ter it ismo ved 2 m. Assume the Qequals 0.25 between bl o ck A an d

the plane and

that the pulley isweightless an d f ricti o nless.


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C lass Problem


T he f ree-b od y d iagram of the pr o blem


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C lass ProblemC lass Problem Energy Energy

H o ckey player S tephan Yelle of Co lo rado Avalanche takes a slapsh o t o ngo al. T he puck leaves his stick at aspee d of 120 km/h an d slid es o n the ice.T he g o alie st o ps the puck with his stick af ter the puck has travele d 12 m.

If the c o eff icient of f ricti o n between the puck an d the ice isQk =0.01 use the law of kinetic energy t o calculate the spee d the puck when it strikes the g o alies stick.

T he ice rink maintenance pers o n acci d entally turns off the icechiller. As a c o nsequence, the ice sur f ace became slushy,increasing the c o eff icient of f ricti o n to 0.10. W hat is thespee d which the puck strike the g o alies stick un d er thisco nd itio n

1)

2)


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H omework

H omework

Pr o blems:

17-3, 17-5, 17-7, 17-9, 17-12, 17-14


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Bonus Problem

Bonus Problem Energy Energy

A spring is use d to sto p a 75 kg package which is m o ving do wn a20 o incline. T he spring has aco nstant k = 25 kN/m an d is hel d

by cables s o that it is initiallyco mpresse d 100mm. I f the vel o cityof

the package is 6 m/s when it is10 m f r o m the spring d etermine themaximum a dd itio nal d efo rmati o nof the spring in bring the package

to

rest. Assume Q = 0.20


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Example Problem Example Problem Energy EnergyDraw the f ree-b od y d iagram of the pr o blem

engr221 lecture 27

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