Lecture 2 Particle Kinematics OK

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Chapter 2. Particle Kinematics

Chapter 2

Particle Kinematics

Chapter Objectives

• To introduce the concepts of position, displacement, velocity,

and acceleration.

• To study particle motion along a straight line and represent this

motion graphically.

• To investigate particle motion along a curved path using

different coordinate systems.

• To present an analysis of dependent motion of two particles.

• To examine the principles of relative motion of two particles

using translating axes.

2.1 Introduction

Kinematics is that branch of dynamics which deals with the description of the motion of

points and rigid bodies without regard for the forces causing that motion. It is purely

descriptive in nature and does not deal with issues of cause and effect. Kinematics is often

referred to as the geometry of motion!. It is based purely upon definitions. "o physical laws,

i.e., observed relations are re#uired. $owever, a thorough wor%ing %nowledge of %inematics

is an absolute prere#uisite to %inetics which is the study of the relationships between motion

the corresponding forces which either cause or accompany the motion.

The study of %inematics will start in this chapter by first discussing the motion of particle.

&ecall that a particle, or a point mass, is a body whose physical dimensions are so small

compared with the radius of curvature of its path. It is an ideali'ed model of a collection of

matter in which the matter is assumed to all be located at a point. $ence we can treat the

motion of the particle as that of a point.

(hen dealing with the motion of points, three %inematical concepts will be of interest. )ach is

represented by a vector*

+i. Position represented by a position vector

+ii. &ate of change of position represented by a velocity vector

+iii. &ate of change of rate of change of position represented by an acceleration vector.

)ach of these will be defined in this chapter and a variety of relations will be derived which

-




can simplify finding expressions for these #uantities. ome of these derived relations will be

restricted to a special motion when the particle is confined to a specified path and the motion

is said to be constrained. The example is the rectilinear motion i.e. straight line motion. If

there are no physical guides, the motion is general in nature, i.e., is not restricted to a

particular type, it is said to be unconstrained.

/arious coordinate systems can be used to describe the position of the particle at any time. It

can be rectangular, cylindrical, spherical coordinate system. The position of a point may also

be described by measurement along the tangent and normal to the curve of the particle path. It

is worth to note that in the case of plane motion where all movement occurs in or can be

represented as occurring in a single plane, the position of a point can be described by only two

coordinate parameters. In three dimensional or spatial cases the number of coordinate

parameters should be three.

0oreover the motion of particles may be described in fixed coordinate system with fixed

reference axes. It this case we provide the absolute motion analysis. (hen we use

coordinates measured from moving reference axes, the relative motion analysis is provided.

2.2 Rectilinear Kinematics

(e will begin our study of dynamics by discussing the %inematics of a particle that moves

along a rectilinear or straight line path. ince a particle has a mass but negligible si'e and

shape we limit application to those ob1ects that have dimensions that are of no conse#uence in

the analysis of the motion. In such problems, the motion of the body is characteri'ed by

motion of its mass center and any rotation of the body is neglected. The %inematics of a

particle is characteri'ed by specifying, at any given instant. the particles position, velocity,

and acceleration.

Consider a particle P moving along a straight line, 3ig 2.4. The path of the particle will be

defined using a single coordinate axis s. The origin O on the path is a fixed point and fromthis point the position vector r is used to specify the location of the particle P at any given

instant t . "otice that r is always, along the s axis, and so its direction never changes. (hat

will change is its magnitude and its sense or arrowhead direction. 3or analytical wor% it is

therefore convenient to represent r by an algebraic scalar s, representing the position

coordinate of the particle. The magnitude of s +and r is the distance from O to P , and the

sense +or arrowhead direction of r is defined by the algebraic sign on s. 5lthough the choice

is arbitrary, in this case s is positive since the coordinate axis is positive to the right of the

origin. 6i%ewise, it is negative if the particle is located to the left of O.

7




The change in the position during the time interval 8t is called the displacement of the

particle. 5t time t+8t the particle has moved from P to P , the displacement is 8r 9 r:r.

;sing algebraic scalars to represent 8r, we also have 8 s 9 s<: s.

3ig. 2.4

"ote that in 3ig.2.4 8 s is positive since the particles final position is to the right of its initial

position, i.e., s = s and the particle moved in the positive s:direction. 6i%ewise, if the final

position is to the left of its initial position, 8 s is negative that means the particle moved in the

negative s:direction. It is worthy to distinguish the displacement which is a vector #uantity,

from the distance the particle travels. The distance traveled is always a positive scalar which

represents the total length of path over which the particle travels.

The average velocity of the particle during the interval 8t is the displacement divided by the

time interval or

∆=

∆avg t

rv +2.4

If we ta%e smaller and smaller values of 8t , the magnitude of 8 s becomes smaller and smaller.

Conse#uently, the average velocity approaches the instantaneous velocity of the particle, or

∆ →

∆=

∆0lim = =

t

d

t dt

r rv r +2.2

&epresenting v as an algebraic scalar, we can also write

= =ds

v sdt

+2.>

Thus the velocity is the time rate of change of the position coordinate s. ince dt is always

positive, the sign used to define the sense of the velocity is the same as that of ds. 3or

example, if the particle is moving to the right, 3ig. 2.4, the velocity is positive? whereas if it is

moving to the left, the velocity is negative. The magnitude of the velocity is %nown as the

speed, and it is generally expressed in units of m/s. @ccasionally, the term Aaverage speedA is

used. The average speed is always a positive scalar and is defined as the total distance +which

is, clearly, a positive scalar, traveled by a particle, sT, divided by the elapsed time 8t , i.e.,

sTB8t .




The average acceleration of the particle during the interval 8t is the change in its velocity

divided by the time interval or

∆= ∆avgt va +2.D

$ere 8v represents the difference in the velocity during the interval 8t . The instantaneous

acceleration at time t is found by ta%ing smaller an smaller values of 8t and corresponding

smaller and smaller values of 8v so the average acceleration approaches the instantaneous

accelerations of the particle, or

∆ →

∆=

∆&

0lim = =

t

d

t dt

v va v +2.E

If using algebraic scalars we get

= =dv

a v dt +2.F

ubstituting )#. +2.> into this result, we can also write

2

2= =

d sa s

dt +2.-

Goth the average and instantaneous acceleration can be either positive or negative. In

particular, when the particle is slowing down , or its speed is decreasing, it is said to be

decelerating. In the case in 3ig. 2.4, if v< + the velocity at P < is less than v, and so 8v 9 v : v

will be negative. Conse#uently, a will also be negative, and therefore it will act to the left , in

the opposite sense to the positive direction and also to v. Clearly, when the velocity is

constant , the acceleration is 'ero. ;nits commonly used to express the magnitude of

acceleration are m/s2.

5 differential e#uation relating the displacement, velocity, and acceleration along the path

may be obtained by eliminating the time differential dt between )#s. +2.> and +2.-. (e geteasily

a ds 9 v dv +2.7

)#s. +2.>, +2.F and +2.7 are the differential e#uations for the rectilinear motion of a particle.

&eali'e that although we can then establish another e#uation, it will not be independent of

)#s. +2.> and +2.F. Keep in mind also that vector and acceleration are actually vector. 3or

rectilinear motion where the direction of the motion is that of the given straight line:path, the

sense of the vector along the path is described by a plus or minus sign. In the treatment or

curvilinear motion we will account for the change in direction of the velocity and

4H




"ow consider the case when the acceleration is given as a function of time, i.e. a=f(t).

ubstitution of the function into )#. +2.F gives

( ) = dv f t dt

+2.42

0ultiplying dt separates the variables and permits integrations. (e get

+ ∫ 0

0

= ( )t

v v f t dt +2.4>

3rom this integrated expression for v as a function of t the position coordinate s is obtained by

integrating )#. +2.>, thus

∫ ∫ 0 0

=

t t

ds vdt

or

+ ∫ 0

0

=t

s s vdt +2.4D

If the indefinite integral is employed the ends condition are used to established the constants

of integration with results which are identical with those obtained by using the definite

integral. If desired, the displacement s may be obtained by a direct solution of second:order

differential e#uation &&= ( )s f t obtained by substitution of f +t into )#. +2.-.

imilarly, when the acceleration is given as a function of velocity a 9 f(v), substitution of the

function into )#. +2.F gives

( ) =dv

f v dt

eparating the variables and integrating yield

= ∫ ∫ 00 t = ( )

t v

v

dv

t d f v +2.4E

This result gives t as a function of v. Then it would be necessary to solve for v as a function of

t so that )#. +2.> can be integrated to obtain the position coordinate s as a function of t .

5lternatively, in this case, the function a 9 f +v can be substituted into )#. +2.7 giving

v dv 9 f +v ds

eparating the variables yields

=∫ ∫ 0 0 ( )

v s

v s

vdv

dsf v

42




Thus

+ ∫ 0

0 =

( )

v

v

vdv s s

f v +2.4F

urely, this form gives the position coordinate s in term of v without explicit reference to t.In the case that acceleration is given as a function of displacement a = f + s , substitution of the

function into )#. +2.7 and integrating give the form

=∫ ∫ 0 0

( )v s

v s

vdv f s ds

Thus we get

= + ∫ 0

2 2

02 ( )

s

s

v v f s ds +2.4-

"ext we solve for v to give v 9 g + s , a function of s. Then we can substitute ds/dt for v,

separate variables, and integrate in the form

=∫ ∫ 0 0

( )

s t

s

dsdt

g s

$ence

= ∫ 0

( )

s

s

dst

gs+2.47

which gives t as a function of s. 6astly we can arrange to obtain the position coordinate s as a

function of t .

In each of the foregoing cases when the acceleration varies according to some functional

relationship, the ability to solve the e#uations by direct mathematical integration will depend

on the form of the function. In cases where the integration is difficult, numerical methods may

be used.

Important Points• ynamics is concerned with bodies that have accelerated motion.

• Kinematics is a study of the geometry of the motion.

• Kinetics is a study of the forces that cause the motion.

• &ectilinear %inematics refers to straight:line motion.

• peed refers to the magnitude of velocity.

• 5verage speed is the total distance traveled divided by the total time. This is

different from the average velocity which is the displacement divided by the

time.

4>




• The acceleration, a 9 dv/dt, is negative when the particle is slowing down or

decelerating.

• 5 particle can have an acceleration and yet have 'ero velocity.

• The relationship it ads = vdv, is derived from a = dv/dt and v= ds/dt, by

eliminating dt.

In summary, the e#uations of rectilinear %inematics should be applied using the following

procedure

Coordinate System

• )stablish a position coordinate s along the path and specify its fixed origin and

positive direction.

• ince motion is along a straight line, the particles position, velocity. and

acceleration can be represented as algebraic scalars. 3or analytical wor% the

sense of s, v, and a is then determined from their algebraic signs .

• The positive sense for each scalar can be indicated by an arrow shown alongside

each %inematic e#uation as it is applied.

Kinematic Euations

• If a relationship is %nown between any two of the four variables a, v, s and t,

then a third variable can be obtained by using one of the %inematic e#uations,

ads = vdv, a = dv/dt and v= ds/dt which relates all three variables.

• (henever integration is performed, it is important that the position and velocity

be %nown at a given instant in order to evaluate either the constant of

integration if an indefinite integral is used, or the limits of integration if a

definite integral is used.

• &emember that )#s. +2. through +2.44 have only a limited use. "ever apply

these e#uations unless it is absolutely certain that the acceleration is constant.

E!"#P$E 2.1

The car in 3ig. 2.2 moves in a straight line such that for a short time its velocity is defined by

v 9 +>t2 J 2t mB s, where t is in seconds. (hen t 9 H, s 9 H. etermine its position and

acceleration when t 9 > seconds.

4D

v




position, velocity, and acceleration. In this section the general aspects of curvilinear motion

are discussed, and in subse#uent sections three types of coordinate systems often used to

analy'e this motion will be introduced.

3ig. 2.> Position

Consider a particle located at point P on a space curve defined by the path function s, 3ig. 2.>.

The position of the particle, measured from a fixed point O, will be designated by the position

vector r = r+t . This vector is a function of time since, in general, both its magnitude and

direction change as the particle moves along the curve.

uppose that during a small time interval 8t the particle moves a distance 8 s along the curve

to a new position P , defined by r' = r + 8r , 3ig. 2.D. The displacement 8r represents the

change in the particles position and is determined by vector subtraction, i.e., 8r=r'-r.

3ig. 2.D isplacement

5gain, during the time 8t. the average velocity of the particle is defined as

avgt

∆=

∆r

v +2.4

The instantaneous velocity of the particle at time t is obtained by choosing shorter and shorter

time interval 8t and correspondingly shorter and shorter vector increment 8r. The

instantaneous velocity is thus represented by the vector

0 lim = =

t

d

t dt ∆ →

∆=

∆r

v r r +2.2H

5s 8t and 8r become shorter it is clear that the direction of 8r approaches the tangent to the

4-

r

P

s

O

r

P

∆ s

Or

P ′

∆r




curve at point P , therefore, the direction of v is also tangent to the curve, 3ig. 2.E. The

magnitude of v, which is called the speed may be obtained by noting that the magnitude of the

displacement 8r is the length of the straight line segment from P to P , 3ig. 2.D.

3ig. 2.E /elocity

&eali'ing that this length, 8r, approaches the arc length 8 s as 8t L H. we have

0 0 lim = lim =t t

r s ds

v st t dt ∆ → ∆ →

∆ ∆= =∆ ∆

Thus. the speed can be obtained by differentiating the with respect to time the length s of the

arc described by the particle.

Consider the velocity v of the particle at time t and also its velocity v< at a later time t J 8t ,

3ig. 2.F. The average acceleration of the particle during the time interval Δt is defined as the

#uotient of 8v and 8t or

avg

t

∆=

∆

va +2.24

where 8v 9 v : v.

3ig.2.F

To study this time rate of change, the two velocity vectors in 3ig. 2.F are plotted in 3ig. 2.-such that their tails are located at the fixed point O and their arrowheads touch points on the

curve. This curve is called a hodograph , and when constructed, it describes the locus of points

for the arrowhead of the velocity vector in the same manner as the path s describes the locus

of points for the arrowhead of the position vector, 3ig. 2.7.

47




3ig. 2.- 3ig. 2.7

This curve is called a hodograph of the motion , and when constructed, it describes the locus

of points for the arrowhead of the velocity vector in the same manner as the path s describes

the locus of points for the arrowhead of the position vector, 3ig. 2.7.

The instantaneous acceleration at time t is obtained by choosing smaller and smaller value for

8t and 8v in )#. +2.24. It is clear that in the limit 8v will approach the tangent to the

hodograph , and so

0 lim = =

t

d

t dt ∆ →

∆=

∆v v

a v& +2.22

5ccording to )#. +2.2H we can also write this result as

2

2

d

dt =

ra +2.2>

Gy definition of the derivative, a acts tangent to the hodograph , 3ig. 2.7, and therefore, in

general, a is not tangent to the path of motion. To clarify this point, reali'e that 8v and

conse#uently a must account for the change made in both the magnitude and direction of the

velocity v as the particle moves from P to P , 3ig. 2.F. Must a magnitude change increases +or

decreases the AlengthA of v, and this in itself would allow a to remain tangent to the path.

$owever, in order for the particle to follow the path, the directional change always AswingsA

the velocity vector toward the AinsideA or Aconcave sideA of the path. and therefore a cannot

remain tangent to the path. In conclusion. v is always tangent to the path and a is always

tangent to the hodograph.

2.( Rectan)ular Components o* +elocity and "cceleration

(hen the position of a particle P is described at any time instant by its rectangular

coordinates x, y and z , it is convenient to resolve the velocity v and the acceleration a of the

particle into rectangular components.

3ig. 2.

4




In such case the motion of a particle is described along a path that is represented using a fixed

x, y, z frame of reference. 5t a given instant the particle P is at point + x, y, z on the curved

path s, 3ig. 2., its location is then defined by the position vector

r 9 xi J y j J z , +2.2D

Gecause of the particle motion and the shape of the path, the x, y, z components of r are

generally all functions of time, i.e., x 9 x+t . y 9 y+t , z 9 z +t , so that r = r+t .

It is easy to determine the magnitude and direction of r. The direction of r is specified by the

components of the unit vector ur 9 rBr and the magnitude of r is defined as

2 2 2 r x y z = + + +2.2E

3ig. 2.4H

The first time derivative of r yields the velocity v of the particle. $ence,

= = ( ) ( ) ( )d d d d

x y z

dt dt dt dt

+ +v r i j k

It is necessary to account for changes in both the magnitude and direction of each of the

vectors components, therefore, the derivative of the i component of v is

( )d dx d

x x dt dt dt

= + i

i i

The second term on the right side is 'ero, since the x, y, z reference frame is fixed , and

therefore the direction +and the magnitude ) of i does not change with time. ifferentiation of

the j and , components may be carried out in a similar manner, hence. we obtain

= =d

x y z dt

+ +v r i j k & & & +2.2F

It follows from this e#uation that the scalar components of the velocity are

, , x y z v x v y v z = = =& & +2.2-

The AdotA notation , , x y z & & represents the first time derivatives of the parametric e#uations

x 9 x+t , y 9 y+t , z= z(t), respectively. 5 positive value for vx indicate that the vector

component vx is directed in the positive direction of i and the sense of each of the other vector

components may be determined in a similar way from the sign of the corresponding scalar

2H




enoting by xH, yH, z 0 the coordinates of the gun, and by +vxH, +vyH, +v'H the components of the

initial velocity vH of the pro1ectile, we integrate twice in t and obtain

0 0 0( ) ( ) ( ) x x y y z z v x v v y v gt v z v = = = = − = =& & &

20 0 0 0 0 01( ) ( ) ( )

2 x y z x x v t y y v t gt z z v t = + = + − = +

If the pro1ectile is fired in the x-y plane from the origin O, we have xH 9 yH 9 z H9H and +v'H 9H,

and the e#uations of motion reduce to

0 0( ) ( ) 0 x x y y z v v v v gt v = = − =

2

0 0

1( ) ( ) 0

2 x y x v t y v t gt z = = − =

These e#uation show that the pro1ectile remains in the x-y plane and that its motion in the

hori'ontal direction is uniform, while its motion in the vertical direction is uniformly

accelerated. Therefore the motion of a pro1ectile may be replaced by two independent

rectilinear motions which are easily visuali'ed if we assume that the pro1ectile is fired

vertically with an initial velocity +vyH from a platform moving with a constant hori'ontal

velocity +vxH. The coordinate x of the pro1ectile is e#ual at any instant to the distance traveled

by the platform, while its coordinate y may be computed as if the pro1ectile were moving

along a vertical line.

It may be observed that the e#uations defining the coordinates x and y of a pro1ectile at any

instant are the parametric e#uations of a parabola. Thus, the tra1ectory of a pro1ectile is

parabolic. This result, however, ceases to be valid when the resistance of the air or the

variation with altitude of the acceleration of gravity is ta%en into account.

Important Points

• Curvilinear motion can cause changes in bt! the magnitude and direction of the

position, velocity, and acceleration vectors.

• The velocity vector is always directed ta"ge"t to the path.

• In general, the acceleration vector is "t tangent to the path, but rather, it is

tangent to the hodograph.

• If the motion is described using rectangular coordinates, then the components

along each of the axes do not change direction, only their magnitude and sense

+algebraic sign will change.

Procedure *or "nalysis

Crd#"ate system

22




+c %%elerat#".

The components of acceleration are determined from )#s. +2.2 and application of the chain

rule. (e have

2

2

(8 ) 0 x x

d t a v x dt = = = =& &&

2 2(2 / 10)

2( ) / 10 2 / 10 2(8) / 10 2(16)(0)/ 10 12.8 y y

d xx a v x x xx m/ s

dt = = = + = + =

&& && &&

Thus

2 2 2 (0) (12.8) 12.8a m/ s= + =

The direction of a is

1 1 o12.8

tan tan 900

y

a x

a

aθ − −

= = =

*te* It is also possible to obtain vy and a y by first expressing y = f +t = +7t 2 B4H 9 F.Dt 2 and

then ta%ing successive time derivatives.

E!"#P$E 2.(

The motion of a box moving along the spiral conveyor shown in 3ig. 2.4> is defined by the

position vector r 9 +H.E sin+2t i J H.E cos+2t j : H.2t , m, where t is in seconds and the

arguments for sine and cosine are in radians +rad . etermine the location of the box whent 9 H.-E s and the magnitudes of its velocity and acceleration at this instant.

3ig. 2.4>

Solution

)valuating r when t 9 H.-E s yields

r - +H.E sin +4.E radi J H.E cos +4.E rad j : H.2+H.-E,

9 +H.Di J H.H>ED j : H.4EH, m

2D




The distance of the box from the origin @ is

2 2 2 (0.499) (0.0354) ( 0.150) 0.522r m= + + − =

The direction of r is obtained from the components of the unit vector ur 9 rBr

ur 9 +H.Di J H. H>ED j H.4EH,/0H.E22 9 H.EEi J H.HF-7 j : H.27-,

$ence, the coordinate direction angles , , , 3ig.2.4>, are

9 cos:4+H.EE 9 4-.2N

9 cos:4 +H. HF-7 9 7F.4o

9 cos:4 +:H.27- 9 4H-N

The velocity is defined by

= = [0.5sin(2 ) 0.5cos(2t) 0.2

= (1cos(2 ) 1sin(2 ) 0.2 )

dt t

dt t t m/ s

+ − =

− −

rv i j k

i j k

$ence, when t 9 H.-E s the magnitude of velocity, or the speed, is

v= O+4 cos +4.E rad 2 J Ol sin +4.E rad 2 J +:H.22 9 4.H2 m/s

The velocity is tangent to the path as shown in 3ig. 2.4>. Its coordinate direction angles can

be determined from uv 9 vBv.

The acceleration a of the box can be determined easily

2

= = [-2sin(2 ) 2cos(2t)

d

t m/ sdt −

v

a i j

It is clear that the acceleration is not tangent to the path.

2. ormal and 3an)ential Components

In previous sections we saw that the velocity of a particle is a vector tangent to the path of the

particle but that, in general, the acceleration is not tangent to the path. It is sometimes

convenient to resolve the acceleration into components directed, respectively, along the

tangent and the normal to the path of the particle. (hen the path along which a particle is

moving is %nown , it is possible to describe the motion using " and t coordinates which act

normal and tangent to the path. respectively, and at the instant considered have their origin

located at the particle.

Planar #otion. (e shall first consider the particle P shown in 3ig.2.4Da which is moving in

a plane along a fixed curve, such that at a given instant it is at position s, measured from point

O. (e attach to the particle a coordinate system that has its origin at a fixed point on the

curve, and at the instant considered this origin happens to coincide with the location of the particle. The t axis is tangent to the curve at P and is positive in the direction of increasing s.

2E




(e will designate this positive direction with the unit vector ut . 5 uni#ue choice for the

normal axis can be made by noting that geometrically the curve is constructed from a series of

differential arc segments, 3ig. 2.4Db. )ach segment ds is formed from the arc of an associated

circle having a radius of curvature and center of curvature O. The normal axis it is

perpendicular to the t axis and is directed from P toward the center of curvature @, 3ig. 2.4Da.

This positive direction, which is always on the concave side of the curve, will be designated

by the unit vector u". The plane which contains the " and t axes is referred to as the osculating

plane , and in this case it is fixed in the plane of motion. The osculating plane may also be

defined as that plane which has the greatest contact with the curve at a point. It is the limiting

position of a plane contacting both the point and the arc segment ds. It is clear that the

osculating plane is always coincident with a plane curve, however. each point on a three:

dimensional curve has a uni#ue osculating plane.

"ow consider the velocity of the particle. ince the particle is moving, s is a function of time.

2F

3ig. 2.4Da 3ig. 2.4Db

3ig. 2.4Dc 3ig. 2.4Dd

3ig. 2.4De 3ig. 2.4Df




5s %nown the particles velocity v has a direction that is always tangent to the path ,

3ig. 2.4Dc , and a magnitude that is determined by ta%ing the time derivative of the path

function s 9 s+t, i.e., v 9 ds/dt . Therefore we may express this vector as the product of the

scalar v and unit vector ut , i.e.

v 9 v ut +2.>4

where v s= .

To obtain the acceleration of the particle we shall differentiate )#. +2.>4 with respect to t .

Thus,

t t v v = +a u u& & +2.>2

In order to determine the time derivative t u& note that as the particle moves along the arc ds in

time dt , ut preserves its magnitude of unity? however. its direction changes, and becomes t ′u .

3ig. 2.4Dd. 5s shown in 3ig. 2.4De. we re#uire t t t d ′ = +u u u . $ere t du stretches between

the arrowheads of t u and t ′u which lie on an infinitesimal arc of radius t 9 4. Therefore, d ut

has a magnitude of dt 9 +4d, and its direction is defined by u". Conse#uently. dut 9 d u",

and therefore the time derivative becomes t nθ =u u&& . ince ds =d, 3ig. 2.4Dd, then

/sθ ρ =& & , and therefore

t n n n

s v θ ρ ρ = = =u u u u

&&&

ubstituting into )#. +2.>2, a can be written as the sum of its two components,

= +t t n na aa u u +2.>>

where t a v = , and

2

n

v a

ρ =

These two mutually perpendicular components are shown in 3ig. 2.4Df in which case the

magnitude of acceleration is the positive value of

= +2 2

t na a a+2.>D

3hree4imensional #otion. If the particle is moving along a space curve, 3ig. 2.4E, then at a

given instant the t axis is uni#uely specified? however, an infinite number of straight lines can

be constructed normal to the tangent axis at P . 5s in the case of planar motion, we will choose

the positive " axis directed from i toward the paths center of curvature O. This axis is

referred to as the principal normal to the curve at P . (ith the " and t axes so defined, )#s.

+2.>4 to +2.>D can be used to determine v and a. ince ut and u" are always perpendicular to

2-




one another : and lie in the osculating plane, for spatial motion a third unit vector, ub defines a

bi:normal axis b which is perpendicular to ut , and un, 3ig. 2.4E.

ince the three unit vectors are related to one another by the vector cross product, e.g.,

ub 9 ut x u"

It may be possible to use this relation to establish the direction of one of the axes, if the

directions of the other two are %nown. 3or example, no motion occurs in the ub direction, and

so if this direction and ut are %nown. then u" can be determined where in this case u" 9 ut x ub.

&emember, though, that u" is always on the concave side of the curve.

3ig. 2.4E

Procedure *or "nalysis

Crd#"ate 1ystem

• Provided the path of the particle is %nown, we can establish a set of " and t

coordinates having a fixed origin which is coincident with the particle at the

instant considered.

• The positive tangent axis acts in the direction of motion and the positive normal

axis is directed toward the paths center of curvature.

• The " and t axes are particularly advantageous for studying the velocity and

acceleration of the particle, because the t and " components of a are expressed by

)#s. +2.>>.

el%#ty

• The particles velocity is always tangent to the path.

• The magnitude of velocity is found from the time derivative of the path function

=v s

2a"ge"t#al %%elerat#"

• The tangential component of acceleration is the result of the time rate of change in

the magnitude of velocity. This component acts in the positive s direction if the

particles speed is increasing or in the opposite direction if the speed is decreasing.

27




• The relations between at , v, t and s are the same as for rectilinear motion, namely,

t a v = , at ds = v dv

• If at is constant , a t 9 +at % the above e#uations, when integrated, yield

2

0 0s = + ( )t cs v t a t 1+ 2

0= + ( )t cv v a t

2 2

0 0= + 2( ) ( )t cv v a s s−

*rmal %%elerat#"

• The normal component of acceleration is the result of the time rate of change in the

direction of the particles velocity. This component is always directed toward the

center of curvature of the path. i.e., along the positive " axis.

• The magnitude of this component is determined from2

=n

v a

ρ

• If the path is expressed as y 9 f + x, the radius of curvature Q at any point on the

path is determined from the e#uation2 3/ 2

2 2

[1 ( / ) =

[ /

dy dx

d y dx ρ

+

The derivation of this result is given in any standard calculus text.

E!"#P$E 2.(

The boxes in 3ig. 2.4Fa travel along the industrial conveyor. If a box as in 3ig. 2.4Fb starts

from rest at and increases its speed such that at 9 +H.2t mBs2, where t is in seconds,

determine the magnitude of its acceleration when it arrives at point .Solution

2

3ig. 2.4Fa 3ig. 2.4Fb




Crd#"ate 1ystem

The position of tile box at any instant is defined from the fixed point using the position or

path coordinate s, 3ig. 2.4Fb. The acceleration is to be determined at , so the origin of the ",

t axes is at this point.

%%elerat#"

To determine the acceleration components =t a v and

2

=n

v a

ρ , it is first necessary to

formulate v and v so that they may be evaluated at . ince v 9 H when t 9 H, then

= 0.2t a v t =& +4

0 0

0.2v t

dv tdt =∫ ∫

20.1v t = +2

The time needed for the box to reach point G can be determined by reali'ing that the position

of G is s 9 > J 2π +2BD 9 F.4D2 m, 3ig. 2.4Fb, and since s 9 H when t 9 H we have

= = 20.1ds

v t dt

=∫ ∫ 6.142

2

0 0

0.1Bt

ds t dt

36.142 0.0333 Bt =

t 9 E.FH s

ubstituting into )#s. +4 and +2 yields

2( ) 0.2(5.690) 1.138 /B t Ba v m s= = =&

2 20.1(5.69) 3.238 /Bv m s= =

5t , 9 2 m, so that

2 2(3.238m/ s)( ) 5.242 /2

BB n

B

v a m s

ρ = = =

The magnitude of a B, 3ig. 2.4Fc, is therefore

2 2 2(1.138) (5.242) 5.36m/sBa = + =

2.5 Cylindrical Components

In some engineering problems it is often convenient to express the path of motion in terms of

cylindrical coordinates, r , , z . If motion is restricted to the plane, the polar coordinates r and

>H

3ig 2.4Fc




are used. These components are called radial and transverse components.

Polar Coordinates. (e can specify the location of particle P shown in 3ig. 2.4-a using both

the radial coordinate* r , which extends outward from the fixed origin O to the particle, and a

transverse coordinate , which is the countercloc%wise angle between a fixed reference line

and the r axis. The angle is generally measured in degrees or radians. (e attach at P two unit

vectors* ur and u which define positive directions of the r and coordinates. $ere ur extends

from P along increasing r , when is held fixed, and u extends from P in a direction that

occurs when r is held fixed and is increased. "ote that these directions are perpendicular to

one another.

3ig. 2.4-

;sing the unit vector ur at any instant the position of the particle is defined by the position

vector

r 9 r ur +2.>E

+elocity. The instantaneous velocity v is obtained by ta%ing the time derivative of r, hence

r r r r = = +v r u u& & & +2.>F

To evaluate &r

u notice thatr

u changes only its direction with respect to time, since by

definition the magnitude of this vector is always unity. $ence, during the time 8t , a change 8r

will not cause a change in the direction of ur , however, a change 8 will cause ur to become

>4

′r

u

ur

8ur

u

8

3ig. 2.4-b3ig. 2.4-a

3ig. 2.4-c

vr

v

3ig. 2.4-d




′r

u where2 = + ∆r r r

u u u , 3ig. 2.4-b. The time change in ur is then 8ur . 3or small angles 8

this vector has a magnitude 8r 9 4+8 and acts in the u direction. Therefore, 8ur =

8 u and we get easily

θ θ =ru u&& +2.>-

ubstituting into the above e#uation for v, the velocity can be written in component form as

r r v v θ θ = = +v r u u& & +2.>7

where

r v r =

v r θ θ =

These components are shown graphically in 3ig. 2.4-c. The radial component vr is a measure

of the rate of increase or decrease in the length of the radial coordinate, i.e., r whereas the

transverse component v can be interpreted as the rate of motion along the circumference of a

circle having a radius r . In particular. the term θ is called the angular velocity. since it

indicates the time rate of change of the angle .

ince vr and v are mutually perpendicular, the magnitude of velocity is simply the positive

value of

2 2( ) ( )v r r θ = + && +2.>

and the direction of v is, of course, tangent to the path at P , 3ig. 2.4-c.

"cceleration. ifferentiating )#. +2.>7 again with respect to time to obtain the acceleration

we write

r r r r r r r θ θ θ θ θ θ = = + + + +a v u u u u u& && && && && & & &

(e can process as before and get the acceleration in component form as

r r a aθ θ = = +a v u u& +2.DH

where

θ = − &&& 2

r a r r

θ θ θ = +&& &2a r r +2.D4

The term θ & #s called the angular acceleration since it measures the change made in the

angular velocity during an instant of time.

Clearly, the magnitude of acceleration is simply the positive value of

2 2 2( ) ( 2 )a r r r r θ θ θ = − + +& && &&& & +2.D2

and the direction is determined from the vector addition of its two components. In general, a

>2




will not be tangent to the path, 3ig. 2.4-d.

Cylindrical Coordinates. If the particle P moves along a space

curve as shown in 3ig. 2.47, then its location may be specified

by the three cylindrical coordinates, r , and z . The z coordinate

is identical to that used for rectangular coordinates. ince the

unit vector defining its direction, u' is constant. the time

derivatives of this vector are 'ero, and therefore the position,

velocity, and acceleration of the particle can be written in terms

of its cylindrical coordinates as follows*

P r z r z = +r u u

r z r r z θ θ = + +v u u u& & +2.D>

2( ) ( 2 )r z r r r r z θ θ θ θ = − + + +a u u u& && &&& & && +2.DD

>>

3ig 2.47

Lecture 2 Particle Kinematics OK

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