ferdinand p. beer systems of particles adapted for ...eng.sut.ac.th/me/box/2_54/425203/ch14 system...
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VECTOR MECHANICS FOR ENGINEERS:
DYNAMICS
Eighth Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
Lecture Notes:
J. Walt Oler
Texas Tech University
CHAPTER
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
14Systems of Particles
Adapted for 425203
Engineering Dynamics
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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14 - 2
Application of Newton’s Laws. Effective Forces• Newton’s second law for each particle Pi
in a system of n particles,
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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14 - 3
Application of Newton’s Laws. Effective Forces
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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14 - 4
Linear & Angular Momentum
LF &rr
=∑ OO HM &rr
=∑
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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14 - 5
Motion of the Mass Center of a System of Particles
• Differentiating twice,
∑
∑
∑
==
==
=
=
=
FLam
Lvmvm
rmrm
G
n
iiiG
n
iiiG
r&rr
rrr
&r
&r
1
1
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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14 - 6
Angular Momentum About the Mass Center
( )
∑
∑
=
′×′=′=
G
n
iiiiG
M
vmrH
r
rrr
1
iGi aaa ′+=rrr
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
Eig
hth
Ed
ition
14 - 7
Angular Momentum About the Mass Center
( )∑=
′×′=′n
iiiiG vmrH
1
rrr
GGi vvv ′+=rrr
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
Eig
hth
Ed
ition
14 - 8
Conservation of Momentum
• If no external forces act on the particles of a system, then the linear momentum and angular momentum about the fixed point O are conserved.
constant constant
00
==
==== ∑∑
O
OO
HL
MHFLrr
r&rr
&r
• In some applications, such as problems involving central forces,
constant constant
00
=≠
==≠= ∑∑
O
OO
HL
MHFLrr
r&rr
&r
• Concept of conservation of momentum also applies to the analysis of the mass center motion,
constant constant
constant
00
==
==
==== ∑∑
GG
G
GG
Hv
vmL
MHFL
rr
rr
r&rr
&r
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
Eig
hth
Ed
ition
14 - 9
Sample Problem 14.2
A 20 N projectile is moving with a velocity of 100 m/s when it explodes into 5 and 15 N fragments. Immediately after the explosion, the fragments travel in the directions θA = 45o and θB = 30o.
Determine the velocity of each fragment.
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
Eig
hth
Ed
ition
14 - 10
Kinetic Energy• Kinetic energy of a system of particles,
( ) ∑∑==
=•=n
iii
n
iiii vmvvmT
1
221
121 rr
iGi vvv ′+=rrr
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
Eig
hth
Ed
ition
14 - 11
Principle of Impulse and Momentum
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
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14 - 12
Sample Problem 14.4
Ball B, of mass mB, is suspended from a cord, of length l, attached to cart A, of mass mA, which can roll freely on a frictionless horizontal tract. While the cart is at rest, the ball is given an initial velocity
Determine (a) the velocity of B as it reaches it maximum elevation, and (b) the maximum vertical distance hthrough which B will rise.
.20 glv =
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
Eig
hth
Ed
ition
14 - 13
Sample Problem 14.4
x
y
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
Eig
hth
Ed
ition
14 - 14
Sample Problem 14.4
© 2007 The McGraw-Hill Companies, Inc. All rights reserved.
Vector Mechanics for Engineers: Dynamics
Eig
hth
Ed
ition
14 - 15
Sample Problem 14.5
Ball A has initial velocity v0 = 10 m /s parallel to the axis of the table. It hits ball B and then ball C which are both at rest. Balls A andC hit the sides of the table squarely at A’ andC’ and ball B hits obliquely at B’.
Assuming perfectly elastic collisions, determine velocities vA, vB, and vC with which the balls hit the sides of the table.