me103 04 work energy
TRANSCRIPT
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MECH103Mechanisms & Dynamics of Machinery
Chapter 4Energy Methods
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Chapter 4 - Principle of Work & Energy• Principle of work & Energy:
– Newton’s second lawdtdm vF
Dot both sides with v and integrate them with respect to t
2 2 2 2 2
1 1 1 1 1
1 ( )2
t t
t t
dd dt m dt m d mddt
r v v
r v v
vF r F v v v v v v
1 [ ( )]2
d dm mdt dt
vF v v v v dt d F v F r
Work done to an object equals the change in the kinetic energy of the object
2
1
)(21)(
21 2
1221122
r
rvvvvrF vvmmdU
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Chapter 4 – Work Done
• Work done by F from r1 to r2
Evaluation of the work
2
1
r
1 2 rU U d F r
s1 s2
Ft
s
2
1
s
s tdsFU
x
y
r1
r2
dr(r1)
z
r2
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Chapter 4 – Work Done
Evaluation of the work
Ft
s1 s2s
2
1
s
s tdsFU
Ft
s1 s2
sNegative work: Motion is in the opposite direction of force (e.g. work done by friction)
Ft
s1 s2s
Constant Force: U = Ft(s2 –s1)
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Chapter 4 – Work Done
Work done by weight
F = - mgj
d dx dy dzr i j k
)( 122
1
2
1
yymgmgdydUy
y
r
rrF
eF rE
rmgRIf 2
2
12
2 11rr
mgRU E
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Chapter 4 – Work Done
Work done by spring
F = k(r – ro)er
21or ( )2 oU k r r
( )o od k x x dx dy dz k x x dxor F r i i j k
F F
ro or xo
F Fr or x
F = k(x – xo)i or
( )o r r od k r r dr rd k r r dr F r e e e
2)(21
oxxkU
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Chapter 4 - Example
• Example 1 – Potential energy of weight
kjiF mg 00
kjir dzdydxd dV = - F dr = mgdz
V2 – V1 = mg(z2 – z1) (= mgh)
2 2 2
1 1 1
V r z
V r zdV d mgdz F r
j
k
i
g
dVd -rF
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Chapter 4 - Example
• Example 2 – Potential energy in gravitational field
rE
rmgR eF 2
2
eer rddrd r
22- E
drdV d mgRr
F r
1
21
11rr
mgRVV E- -1 1
22
V r
EV r
drdV mgRr
y
x
r
mdr
drer
rde
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PowerRate at which work is done
Unit: Joule/sec or Watt, hp (=746 Watt)=550 ft-lb/s
2
21 mv
dtd
dtdUP
Average Power:
2
1
2 22 1
2 1 2 1
1 12
t
av t
mP Pdt v vt t t t
http://www.sizes.com/units/horsepower_british.htm
Dynamometer: Engine horsepower measurements, transmission fatigue testing, and driveline parasitic horsepower loss measurements.
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dVd -rF If there exists a function V of position only such that
2
1
r 2 21 2 1 2 2 1r
1 ( )2
U d V V m v v F r
222
211 2
121 mvVmvV 21i.e. Constant
2V mv
V(x,y,z) is called the potential energy.
Principle of Conservation of Energy The sum of the kinetic energy and potential
energy (or strain energy) is constant. True only if F is conservative!!
: kinetic energy
2211 VTVT
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Chapter 4 – Example - Pendulum
• Example 3 – Conservation of energy : kinetic energyV : potential energy
21At point A, 2Ah L v gL v
1 1 2 2T V T V
2 21 1 2 2
1 12 2
mv mgy mv mgy
2 22 1 2 1 12 ( ) [ 2 ]v g y y v gh v
v1mg
y
x
mg
FL
v2
LF
v1 L
v3
mg
F
y1
y3
mg
FL
v2
y2
h
Amg
FL
v2
y2
sinLh
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Example 3-6: Pendulum
The mass m is released from rest with the string horizontal. By using Newton’s 2nd law in terms of polar coordinates, determine the velocity magnitude v and the tension T in the string as functions of
cos2
2
mgdtdmr
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Ex 3-6: Pendulum
Solution Draw free body diagram Express the force in polar coordinates
Weight: ( sin cos )rmg W e e
2
2
Acceleration: 2
0
r
r
r r r r
L L r, r
a e e
e e
Tension : rTT e
m T W a
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2 2sinrF T mg mL mL
cosF mg mL
21 sin , (for 0 at 0) or2
2 sin
gL
v L gL
cosd gd L
2sinrF T mg mL sin3sinsin2 mgmgmgT
Ex 3-6: Pendulum
How to solve this problem using Energy Method?
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Chapter 4 – EnergyPotential energy (strain energy) of linear spring
or F = k(r – ro)er
21or ( )2 oU k r r
( ) o o
dV dU dk x x dx dy dz k x x dx or
F ri i j k
F F
ro or xo
F Fr or x
F = k(x – xo)i
( )o r r od k r r dr rd k r r dr F r e e e
2
Strain energy1 ( )2 oU k x x
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Example 4-1 – Potential energy (strain energy) : spring and friction
A spring is used to stop a 60-kg package which is
sliding on a surface. The spring has a constant
k=20kN/m and is held by cables so that it is initially
compressed 120mm. Knowing that the package has
v= 2.5m/s shown and that the max additional spring
deflection is 40 mm, determine (a) the coefficient of
kinetic friction (k) between the package and the
surface, (b) the vel. of the package as it passes
again through the position shown
S.E. = Strain Energy: one kind of potential energy
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Ex 4-1 – Example-package stopped by spring
• Solution Free Body Diagram
k=0
W
Nf = kN
F= kx
Energy in each state
1
v1x0 +x1
x1 : Initial compression
21
21
1. .21. .2
K E mv
S E k x
State 1Initial Condition
2s
v2 = 0
x0 + x2
22
22
1. . 021. .2
K E mv
S E k x
State 2Spring is fully compressed
x2 : Final compressions : total distance traveled
2 2 21 1 2
1 1 12 2 2
mv k x k x
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Conservation of energyInitial K.E. of the package – |Work done by friction|+ initial S.E. stored in spring = Zero K.E. + S.E. stored in spring
21
21
1. .21. .2
K E mv
S E k x
State 1Initial Condition
22
22
1. . 021. .2
K E mv
S E k x
State 2Spring is fully compressed
2 2 21 1 2
1 1 12 2 2kmv mg s k x k x k is unknown
1
v1 v2 = 0
2s
2 1 0.04 0.16x x m m 1 0.12x m
2 10.6 .ks x x can be determined 1 2.5 /v m s
Ex 4-1 – Example-package stopped by spring
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2v2 = 0
22
22
1. . 021. .2
K E mv
S E k x
State 2Spring is fully compressed
3
v3
23
21
1. .21. .2
K E mv
S E k x
State 3Rebounce
s2
v2 = 0
W
N f = kN
F=kx
Conservation of energyFinal K.E. of the package (State 3) + Initial S.E. stored in spring (State 3 = State1) = S.E. stored in spring (State 2)– Work done by friction
2 2 23 1 2
1 1 12 2 2 kmv k x k x mg s v3 is unknown
2 1 0.04x x m 1 0.12x m 2 10.6s x x
3 .v can be determined
Ex 4-1 – Example-package stopped by spring
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Example 4-2: Crate on inclined surfaceThe two crates in the figure are released from rest. Their masses are mA=40 kg and mB = 30 kg, and the kinetic coefficient of friction between crate A and the inclined surface is k = 0.15. What is their velocity when they have moved 400mm?
v
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• Solution Free Body Diagram mA g
Nf = kNT = 20
T
mB g
Crate A Crate B
Method 1:Consider Crates A & B separately
BAivvmdsΣF i
s
s t ,),(21 2
122
2
1
)0(
21)20cos(20sin 24.0
0 vmdsgmgmT AAkA
Crate A
Crate B0.4 2
0
1( ) ( 0)2B Bm g T ds m v 2 eqs, 2 unknowns
BA
kBA
mmmm
T)sincos1(
BA
kAB
mmmmsg
v
)cos(sin(22 v = 2.07 m/s
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Conservation of energy
0
21
21
)20cos(20sin
22
4.0
0
4.0
0
vmvm
gdsmdsgmgm
BA
BAkA
.determinedbecanv
Example 4-2: Crate on inclined surface
Method 2:Consider Crates A & B together
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Ex 4-3: Car on the track
A 2000 lb car starts from rest at point 1 and
moves w/o friction down the track shown. (a)
Determine the force exerted by the track on
the car at point 2, where the radius of
curvature of the track is 20 ft. (b) Determine
the minimum safe value of the radius of
curvature at point 3.
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Ex 4-3: at point 2
• Solution
W
N
man = mv22/
State 2
Conservation of energy2 21 1 2 2
1 12 2
mv mgh mv mgh
and can be determinednN a .
22
nvN mg ma m
1 1 2
2
0, 40 .
v h h ftv can be determined.
Centripetal acceleration
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Ex 4-3: at point 3
W
man = mv32/N
State 3
Conservation of energy
2 21 1 3 3
1 12 2
mv mgh mv mgh
23
nvN mg ma m
Min. value of can be determined by setting 0N .
1 1 3
3
0, 25 .
v h h ftv can be determined.
Centripetal acceleration
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Ex 4-3: Skier on the ramp
Textbook Ex 15.4
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Ex 4-3: Skier on the ramp
222
211 2
121 mvmgymvmgy
Textbook Ex 15.4
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Solution On the top of ramp (State 1); leaving the ramp (State 2)
2 21 1 2 2
1 12 2
mv mgh mv mgh
Add (jump) vertical velocity of 3 m/s at point 2
22
22 3 vv
1 1 2
2
0, 20 (given)the horizontal component can be determined
v h h meterv .
Ex 4-3: Skier on the ramp
v2 = 24.8 m/s
v’2 = 25.0 m/s
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Leaving the ramp (State 2) to the highest point (State 3)
2 22 2 3 3
1 12 2
m v mgh mv mgh
2 2 22 2
3 2
3
3 .At the highest point of jump:
can be determined
v vv v
h .
Ex 4-3: Skier on the ramp
h3 = 0.459 m
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Example 4.4 Net Work by Internal Forces
• QuestionCrates A and B in Fig are released from rest. The coefficient of kinetic friction between A and B is μk, and the friction between B and the inclined surface can be neglected. What is the velocity of the crates when they have moved a distance b?
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Example 4.4 Net Work by Internal Forces
• StrategyBy applying the principle of work and energy to each crate, we can obtain two equations in terms of the tension in the cable and the velocity.
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Example 4.4 Net Work by Internal Forces
• SolutionWe draw the free-body diagrams of the crates in Figs a & b. The acceleration of A normal to the inclined surface is zero, so N = mAgcos θ. The magnitudes of the velocities of A and B are equal (Fig c). The work done on A equals the change in its kinetic energy.
1
21cossin
21
220
21
2212
vmdsgmgmT
vvmU
Ab
AkA
A
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Example 4.4 Net Work by Internal Forces
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Example 4.4 Net Work by Internal Forces
The work done on B equals the change in its kinetic energy.
221cossin
21
220
21
2212
vmdsgmgmT
vvmU
Bb
AkB
B
Summing Eqns (1) & (2) to eliminate T to solvefor v2, we obtain
BAAkAB mmmmmgbv cos2sin22