2-Chapters ME201 071 Chapter12

7/27/2019 2-Chapters ME201 071 Chapter12

1/9

Kinematics of a Particle

Introduction

Mechanics is a branch of the physical sciences that is concerned with the state of rest or motion of bodies

subjected to the action of forces.

o Statics is concerned with the equilibrium of a body that is either at rest or moves with constant velocity.

o Dynamics deals with the accelerated motion of a body.

Kinematics is a study of the geometry of the motion of a body. (i.e It defines the relationship

among displacement, velocity and acceleration of a moving body)

Kinetics is a study of the forces causing the motion of a body. (i.e It defines the relationshipbetween the forces that act on a body and the motion of the body)

Classification of Motion Translatory motion or rectilinear motion or straight line motion.

Curvilinear motion Rotary motion or rotational motion

Rectilinear Kinematics: Continuous Motion

Rectilinear kinematics refers to straight-line motion.

The kinematics of a particle is characterized by specifying the particles position, velocity, and acceleration

at any given instant.

A car moving in a straight road is an example of rectilinear motion.

Mechanics

Statics Dynamics

Kinematics Kinetics


2/9

Distance and Displacement:

Distance and displacement are two quantities that may seem to mean the same thing yet have distinctly

different definitions and meanings.

Displacement is a vector quantity, that refers to the object's overall change in position. Displacement is

a measurement of change in position of the particle in motion. Its magnitude and direction are measured by

the length and direction of the straight line joining initial and final positions of the particle. Obviously, the

length of the straight line between the positions is the shortest distance between the points.

Distance is a scalar quantity that refers to "how much ground an object has covered" during its

motion.

Velocity:

o If the particle moves through a displacement r during a time interval t, the average velocity of the

particle during this time interval is

t

avg

=r

v

o The instantaneous velocity is the rate of change of displacement with time. It is expressed as

dt

dsv = (1)

o Note that velocity is a vector quantity. It is different from speed, which is distance divided by time.

Acceleration:

o The average acceleration of the particle is

tavg

=v

a (i.e increase in magnitude of velocity with respect to time )

o The instantaneous acceleration is the rate of change of velocity with time. It is expressed as

dt

dva = ( i.e a = dv/ds * ds/dt ) (2)

o Substituting equation (1) into (2) results in

2

2

dt

sda = (3)
http://1.bp.blogspot.com/-7SRZ4rEk-mI/TbezLP5woAI/AAAAAAAAAJM/RBi6Gs84TRo/s1600/rectilinear-motion-of-object.jpeg


3/9

o Note that acceleration is a vector quantity.

o When the particle is slowing down, it is said to be decelerating. In this case, acceleration will be

negative (i.e. the body is accelerating in the direction opposite to the direction of v).

o When the velocity is constant, the acceleration is zero .

A differential relation involving the displacement, velocity, and acceleration along the path may be

obtained by eliminating dtbetween equations (1) and (2). The result is

dvvdsa = (4)

Constant Acceleration, a = ac

Consider a particle with the following initial conditions:

s =so and v = vo when t= 0.

When the acceleration is constant, equations (1), (2), and (4) can be integrated to obtain the following:

.. (5)

.. (6)

.. (7)

12.4 General Curvilinear Motion

Curvilinear motion occurs when the particle moves along a curved path.

For curvilinear motion, the particles position, velocity, and acceleration are formulated by using vectoranalysis.

Consider a particle initially located at point P. During the time interval t, the particle moves a distance salong a curve defined by the path function s .

o The initial position of the particle, measured from a fixed point O, is designated by the position vector r

= r(t).

o The displacement during the time interval is .rrr =

o The instantaneous velocity is

dt

drv = (8)

Note: The direction ofv is tangent to the curve.

=

2

2

1tatvss

coo++=

( )ocossavv += 2

22


4/9

o The instantaneous speed is

dt

dsv = (9)

Thus, the speed can be obtained by differentiating the path functions with respect to time.

o The locus of points for the arrowheads of the velocity vector forms a curve called the hodograph (see

details on page 32 of textbook).

o The instantaneous acceleration is

dt

dva = (10)

o Substituting equation (9) into (10) results in

2

2

dt

d sa = (11)

o The acceleration acts tangent to the hodograph.

12.5 Curvilinear Motion: Rectangular Components

The motion of a particle can be described along a path that is represented using a fixed x, y, z frame of

reference.

Consider a particle at point (x, y, z) on the curved path s shown in fig. 12-17.

Position:

o The location of the particle is defined by the position vector

r = xi + yj + zk (12)

where x, y, z are the components ofr and they are functions of time.

o The magnitude ofr is

222 zyxr ++=

o The direction ofr is specified by the components of the unit vectorur = r/r.

Velocity:

o The velocity of the particle is

kjir

v zyx vvvdt

d++== (13)

where the components ofv are the first time derivatives of x, y, z, i.e.

zvyvxv zyx === (14)

o The magnitude ofv is

222

zyx vvvv ++=


5/9

o The direction ofv is specified by the components of the unit vector uv = v/v. This direction is always

tangent to the path (see fig. 12-17b).

Acceleration:

o The acceleration of the particle is

kji

va

zyx

aaadt

d++==

(15)

where

zva

yva

xva

zz

yy

xx

==

==

==

(16)

o The magnitude ofa is

222 zyx aaaa ++=

o The direction ofa is specified by the components of the unit vectorua = a/a. In general, the direction is

not tangent to the path; it is tangent to the hodograph (see fig. 12-17c).

Note: Since rectilinear motion occurs along each coordinate axis, the motion of each component is found using

equations (1), (2), and (4).

12.6 Motion of a Projectile

A projectile is an object that is thrown or is fired from a weapon (e.g. a bullet, stone).

The motion of a projectile is often studied in terms of its rectangular components.

Consider a projectile launched at point (xo, yo), as shown in Fig. 12-20 of the course textbook. The path is

defined in the x-y plane and the initial velocity has components (vo)x and (vo)y. Neglecting air resistance, thehorizontal and vertical motions are described as follows:

o Horizontal Motion (ax = 0):

( ) ( ) tvxxvvxooxox

+== (a)

o Vertical Motion (ay = -g):

Equations (1), (2), and (4) give

( ) tgvv yoy = (b)

( )2

2

1tgtvyy

yoo+= (c)

( ) ( )oyoy yygvv = 222

(d)

Note: By eliminating t between equations (b) and (c), equation (d) is obtained. Therefore, only two ofequations (b), (c), (d) are independent.


6/9

The problem involving projectile motion can have at most 3 unknowns. The 3 equations to be used areequation (a), and two of equations (b), (c), and (d).

12.7 Curvilinear Motion: Normal and Tangential Components

When the path along which a particle is moving is known, it is often convenient to describe the motionusing n and t coordinates which act normal and tangent to the path, respectively, and at the instantconsidered have their origin located at the particle.

Planar Motion

Consider a particlePwhich is moving in a plane along a fixed curve as shown in Fig. 12-24.

o A coordinate system whose origin coincides with the location of the particle is considered (i.e. it moves

with the particle).

o The taxis is tangent to the curve atPand the n axis is normal to the curve atP.

o The unit vector on the taxis is designated as ut. It is positive in the direction of motion.

o The unit vector on the n axis is designated as un. It is directed toward the center of curvature of the path

(see Fig. 12-24a).

Velocity

The position of the particle from a fixed point O (Fig. 12-24) iss. Note thats =s(t).

The particles velocity v is always tangent to the path. Hence,

tvuv=

(17)

where

sdtdsv == (18)

Acceleration

The acceleration of the particle has components on the two axes. It is written as

nntt aa uua += (19)

where

vat = or dvvdsat = (20)

and

2v

an = (21)

In equation (21), is the radius of curvature of the path at P. If the path is expressed asy =f(x), at any

point on the path is determined from the equation


7/9

( )[ ]

22

2/321

dxyd

dxdy+= (22)

The tangential component of acceleration, at, is the result of the time rate of change in the magnitude of

velocity. The relations between at, v, t and s (i.e. equations 20) are the same as for rectilinear motion. It at is

constant, equations 5-7 apply.

The normal component of acceleration, an, is often referred to as the centripetal acceleration.

The magnitude of the acceleration is

22 nt aaa += (23)

Special cases

1. If the particle moves along a straight line, then and, therefore, an =0. Thus, vaa t == .

2. If the particle moves along a curve with a constant speed, then 0==vat and a = an = v2/.

12.8 Curvilinear Motion: Cylindrical Components

For some engineering problems, it is convenient to express the path of motion in terms of cylindrical

coordinates, r, ,z.

If the motion is restricted to the plane, the polar coordinates rand are used.

Polar Coordinates

The radial coordinate, r, extends outward from the fixed origin O to the particle. The unit vectorur defines

the positive direction of the rcoordinate (see Fig. 12-30a). The transverse coordinate, , is the counterclockwise angle between a fixed reference line and the raxis

The unit vectoru defines the positive direction of the coordinate (see Fig. 12-30a). The angle is generallymeasured in degrees or radians, where 1 rad = 180o/.

Position

o At any instant the position of the particle is defined by the position vector

rrur = (24)

Velocityo The instantaneous velocity of the particle is

uuv vv rr += (25)

where

rv

rvr

=

=

(26)

o The radial component vr is a measure of the rate of increase or decrease in the length of the radial

coordinate.


8/9

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

having a radius r. The term is called the angular velocity.

o The magnitude of velocity (i.e. speed) is

22 vvv r += (27)

o The direction of the velocity is tangent to the path (see Fig. 12-30c).

Acceleration

o The instantaneous acceleration of the particle is

uua aa rr += (28)

where

rra

rrar

2

2

+=

=

(29)

o The term 22 dtd = is called the angular acceleration.

o The magnitude of acceleration is

22 aaa r += (30)

Cylindrical Coordinates

If the particle moves along a space curve as shown in Fig. 12-31, then its location may be specified by the

three cylindrical coordinates.

The position, velocity, and acceleration of the particle can be written in terms of its cylindrical coordinates

as follows:

zr zr uur +=

zr zrr uuuv ++=

zr

zrrrr uuua +++=

22

Time Derivatives

The equations of kinematics requires that we obtain the time derivatives ,r ,r , and in order toevaluate the rand components ofv and a. See pages 65 and 66 of the textbook for examples in which

these derivatives are obtained.

12.9 Absolute Dependent Motion Analysis of Two Particles

In some types of problems the motion of one particle will depend on the corresponding motion of another

particle. For example, in Fig. 12-36 of the textbook, the blocks A and B connected with a cord have their

motions depending on each other.


9/9

Analysis of the motions involves developing equation for length of cord as a function of the positioncoordinates of the particles.

The procedure for analysis is itemized on page 78 of the textbook.

12.10 Relative-Motion Analysis of Two Particles Using Translating Axes

So far, we have determined the motion of a particle using a single fixed reference frame.

When the path of motion for a particle is complicated, it may be feasible to analyze the motion in parts byusing two or more frames of reference.

In this section, the use of two frames of reference is considered. One frame is fixed while the other one is

translating.

Consider particles A and B moving along the arbitrary paths aa and bb, respectively, as shown in Fig. 12-

42a.

o The absolute positions of the particles, measured from the fixed origin O, are rA and rB.

o

The relative position of B with respect to A is represented by the vectorrB/A.o The three vectors can be related by the equation

rB = rA + rB/A (31)

o Similarly, the velocity and acceleration vectors are related by

vB = vA + vB/A (32)

aB = aA + aB/A (33)

(Study examples 12.25 12.27.)

Documents

2-Chapters ME201 071 Chapter12