VECTOR MECHANICS FOR ENGINEERS:

DYNAMICS

Eighth Edition

Ferdinand P. Beer

E. Russell Johnston, Jr.

Lecture Notes:

J. Walt Oler

Texas Tech University

CHAPTER

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14Systems of Particles

Adapted for 425203

Engineering Dynamics

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Vector Mechanics for Engineers: Dynamics

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Application of Newton’s Laws. Effective Forces• Newton’s second law for each particle Pi

in a system of n particles,

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Application of Newton’s Laws. Effective Forces

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Linear & Angular Momentum

LF &rr

=∑ OO HM &rr

=∑

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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

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Angular Momentum About the Mass Center

( )

∑

∑

=

′×′=′=

G

n

iiiiG

M

vmrH

r

rrr

1

iGi aaa ′+=rrr

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Angular Momentum About the Mass Center

( )∑=

′×′=′n

iiiiG vmrH

1

rrr

GGi vvv ′+=rrr

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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

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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.

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Kinetic Energy• Kinetic energy of a system of particles,

( ) ∑∑==

=•=n

iii

n

iiii vmvvmT

1

221

121 rr

iGi vvv ′+=rrr

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Principle of Impulse and Momentum

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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 =

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Sample Problem 14.4

x

y

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Sample Problem 14.4

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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.