# dyna - beer

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Chapter 11 KINEMATICS OF PARTICLES

x

PO

x

The motion of a particle along astraight line is termed rectilinearmotion. To define the positionPof the particle on that line, we

choose a fixed origin O and apositive direction. The distancex from O toP, with the

appropriate sign, completely defines the position of the particle

on the line and is called the position coordinateof theparticle.

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x

PO

x

The velocity v of the particle is equal to the time derivative ofthe position coordinatex,

v =dx

dtand the accelerationa is obtained by differentiating v withrespect to t,

a =dv

dt

or a =d2x

dt2

we can also express a as

a = vdv

dx

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x

PO

x

v =dx

dta =

dv

dt

or a = d2

xdt2 a = v dvdxor

The velocity v and acceleration a are represented by algebraic

numbers which can be positive or negative. A positive value for

v indicates that the particle moves in the positive direction, anda negative value that it moves in the negative direction. A

positive value fora, however, may mean that the particle is truly

accelerated (i.e., moves faster) in the positive direction, or that

it is decelerated (i.e., moves more slowly) in the negativedirection. A negative value fora is subject to a similar

interpretation.

+-

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Two types of motion are frequently encountered: uniform

rectilinear motion, in which the velocity v of the particle is

constant and

x =xo + vt

and uniformly accelerated rectilinear motion, in which the

acceleration a of the particle is constant and

v = vo + at

x =xo

+ vot + at2

1

2

v2 = vo + 2a(x -xo )2

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x

O

xA

xB

xB/A

A B

When particlesA andB move along the same straight line, the

relative motion ofB with respect toA can be considered.

Denoting byxB/Athe relative position coordinate ofB with respect

toA , we have

xB =xA +xB/A

Differentiating twice with respect to t, we obtain

vB = vA + vB/A aB = aA + aB/A

where vB/A and aB/A represent, respectively, the relative velocityand the relative acceleration ofB with respect toA.

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A

B

C

xA

xB

xC

When several blocks are are connected by inextensible cords,

it is possible to write a linear relation between their position

coordinates. Similar relations can then be written between

their velocities and their accelerations and can be used toanalyze their motion.

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Sometimes it is convenient to use a graphical solution for

problems involving rectilinear motion of a particle. The graphical

solution most commonly involvesx - t, v - t, and a - tcurves.

a

tv

tx

t

t1 t2

v1

v2

t1 t2

v2 - v1 = a dtt1

t2

x1

x2

t1 t2

x2 -x1 = v dtt1

t2

At any given time t,

v = slope ofx - tcurve

a = slope ofv - tcurve

while over any given time interval

t1 to t2,

v2 - v1 = area undera - tcurve

x2 -x1 = area underv - tcurve

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x

y

r

P

Po

O

v

s

The curvilinear motion of a particle

involves particle motion along a

curved path. The positionPof the

particle at a given time is definedby theposition vectorr joining the

origin O of the coordinate system

with the pointP.

The velocity v of the particle is defined by the relation

v =dr

dtThe velocity vector is tangent to the path of the particle, and

has a magnitude v equal to the time derivative of the lengths ofthe arc described by the particle:

v =ds

dt

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x

y

r

P

Po

O

v

s

v =dr

dt

In general, the acceleration aof the particle is not tangent

to the path of the particle. It

is defined by the relation

v =ds

dt

a =dv

dtx

y

r P

Po

O

a

s

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x

y

zi

j

k

vx

vy

vz

xiyj

zk

P

x

y

z

i

j

k

r

ax

ay

az

P

Denoting byx,y, andzthe rectangular

coordinates of a particleP, the

rectangular components of velocity and

acceleration ofPare equal, respectively,to the first and second derivatives with

respect to tof the corresponding

coordinates:

vx =x vy =y vz=z. . .

ax =x ay =y az=z.. .. ..

r

The use of rectangular components is

particularly effective in the study of the

motion of projectiles.

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x

y

z

x

y

z

A

B

rA

rB rB/A

For two particles A andB moving

in space, we consider the

relative motion ofB with respect

toA , or more precisely, withrespect to a moving frame

attached toA and in translation

withA. Denoting by rB/Athe

relative position vector ofB with

respect toA , we have

rB = rA + rB/A

Denoting by vB/Aand aB/A, respectively, the relative velocityand

the relative acceleration ofB with respect toA, we also have

vB = vA + vB/A

aB = aA + aB/Aand

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x

y

C

P

an = en

O

v 2

r

at = etdv

dt

It is sometimes convenient to

resolve the velocity and acceleration

of a particlePinto components other

than the rectangularx,y, andz

components. For a particlePmoving

along a path confined to a plane, we

attach toP the unit vectors et

tangent to the path and en normal to

the path and directed toward thecenter of curvature of the path.

The velocity and acceleration are expressed in terms of tangential

and normal components. The velocity of the particle is

v = vet

The acceleration is

a = et + env2

r

dv

dt

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v = vet

In these equations, v is the speed of the particle and r is theradius of curvature of its path. The velocity vectorv is directed

along the tangent to the path. The acceleration vectora

consists of a component at directed along the tangent to thepath and a component an directed toward the center of

curvature of the path,

a = et + env2r

dvdt

x

y

C

P

an = en

O

v 2

r

at = etdv

dt

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x

P

O

eq

q

r = rer

erWhen the position of a particle moving

in a plane is defined by its polar

coordinates r and q, it is convenient to

use radial and transverse componentsdirected, respectively, along the

position vectorr of the particle and in

the direction obtained by rotating r

through 90

o

counterclockwise. Unitvectors er and eq are attached toPand are directed in the radial

and transverse directions. The velocity and acceleration of the

particle in terms of radial and transverse components is

v = rer+ rqeq. .

a = (r- rq2)er + (rq + 2rq)eq... .. . .

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x

P

O

eq

q

r = rer

erv = rer+ rqeq

. .

a = (r- rq2)er + (rq + 2rq)eq... .. . .

In these equations the dots represent differentiation withrespect to time. The scalar components of of the velocity

and acceleration in the radial and transverse directions are

therefore

vr= r vq= rq. .

ar= r- rq2 aq = rq + 2rq... .. . .

It is important to note that ar is not equal to the time derivative

ofvr, and that aq is not equal to the time derivative ofvq.

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Chapter 12 KINETICS OF PARTICLES:

NEWTONS SECOND LAW

Denoting by m the mass of a particle, by S F the sum, orresultant, of the forces acting on the particle, and by a the

acceleration of the particle relative to a newtonian frame of

reference, we write

S F = maIntroducing the linear momentum of a particle,L = mv,

Newtons second law can also be written as

S F = L.

which expresses that the resultant of the forces acting on a

particle is equal to the rate of change of the linear momentum

of the particle.

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To solve a problem involving the motion of a

particle, S F = ma should be replaced byequations containing scalar quantities. Using

rectangular components ofF and a, we have

SFx = max SFy= may SFz= maz

x

y

P

an

O

at

x

y

z

ax

ay

az

P

x

P

aq

O

ar

Using tangential and normal components,

SFt = mat= m dvdtv2

rUsing radial and transverse components,

...

.. . .SFr = mar= m(r- rq2)

qr

SFn = man= m

SFq = maq= m(rq + 2rq)

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