Chapter 11 : Kinematics of Particles -...

Chapter 11

Mohammad Suliman Abuhaiba,Ph.D., P.E.1

Kinematics of Particles

2/1/2017 10:00 AM

First Exam

Saturday

18/2/2017

Mohammad Suliman Abuhaiba,Ph.D., P.E.

2 2/1/2017 10:00 AM

IntroductionMechanics

Mechanics = science which

describes and predicts the

conditions of rest or motion of

bodies under the action of forces

It is divided into three parts:

1. Mechanics of rigid bodies

2. Mechanics of deformable bodies

3. Mechanics of fluidsMohammad Suliman Abuhaiba,Ph.D., P.E.

3 2/1/2017 10:00 AM

Introduction

Mechanics of rigid bodies issubdivided into:

1. Statics: deals with bodies at rest

2. Dynamics: deals with bodies in

motion


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Introduction

Dynamics is subdivided into:

1. Kinematics

study of geometry of motion

relating displacement, velocity,

acceleration, and time without referenceto the cause of motion

2. Kinetics

study of the relation existing between the

forces acting on a body, the mass of thebody, and the motion of the body


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Introduction

A dynamic study could be done ontwo levels:

1. Particle

an object whose size and shape can

be ignored when studying its motion.

2. Rigid Body

a collection of particles that remain at

fixed distance from each other at all

times and under all conditions of

loading. Mohammad Suliman Abuhaiba,Ph.D., P.E.

6 2/1/2017 10:00 AM

Motion of Particles

1. Rectilinear Motion

2. Curvilinear Motion


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Rectilinear Motion of Particles

Position


8

Velocity

t

xv

t

x

t

0lim

Average velocity

Instantaneous

velocity

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AccelerationMohammad Suliman Abuhaiba,Ph.D., P.E.

9

Instantaneous

acceleration t

va

t

0lim

t

v

Average acceleration

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10

• Consider particle with motion given by

326 ttx

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Three classes of motion may be defined:

1.Acceleration is a function of time, a = f(t)

2.Acceleration is a function of position, a = f(x)

3.Acceleration is a function of velocity, a = f(v)

Determination of Motion of a Particle


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12

1. Acceleration is a function of time, a = f(t)

tttx

x

tttv

v

dttvxtxdttvdx

dttvdxtvdt

dx

dttfvtvdttfdv

dttfdvtfadt

dv

0

0

0

0

0

0

0

0

2/1/2017 10:00 AM



13

2. Acceleration is a function of position, a = f(x)

x

x

x

x

xv

v

dxxfvxv

dxxfdvvdxxfdvv

xfdx

dvva

dt

dva

v

dxdt

dt

dxv

0

00

2

0212

21

or or

2/1/2017 10:00 AM



14

3. Acceleration is a function of velocity, a = f(v)

tv

v

tv

v

tx

x

tv

v

ttv

v

vf

dvvxtx

vf

dvvdx

vf

dvvdxvfa

dx

dvv

tvf

dvdt

vf

dv

dtvf

dvvfa

dt

dv

000

00

0

0

2/1/2017 10:00 AM

Sample 11.2


15

Determine:

a. velocity & elevation above

ground at time t

b. highest elevation reached by

ball and corresponding time

c. time when ball will hit the

ground & corresponding

velocity

Ball tossed with 10 m/s vertical

velocity from window 20 m

above ground.

2/1/2017 10:00 AM

Sample 11.3


16

Brake mechanism used to reduce gun recoil consists

of piston attached to barrel moving in fixed cylinder

filled with oil. As barrel recoils with initial velocity

v0, piston moves and oil is forced through orifices in

piston, causing piston and cylinder to decelerate at

rate proportional to their velocity; a = -kv

Determine v(t), x(t), and v(x).

2/1/2017 10:00 AM

Assignment #11.1

1, 6, 11, 17, 22, 29


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Uniform Rectilinear Motion


18

Acceleration is zero and velocity is constant

vtxx

vtxx

dtvdx

vdt

dx

tx

x

0

0

00

constant

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Uniformly Accelerated Rectilinear Motion


19

Acceleration of the particle is constant

atvv

atvvdtadvadt

dv tv

v

0

000

constant

221

00

221

000

00

0

attvxx

attvxxdtatvdxatvdt

dx tx

x

020

2

020

221

2

constant

00

xxavv

xxavvdxadvvadx

dvv

x

x

v

v

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Motion of Several Particles Relative Motion


20

ABAB xxx relative position of B wrt A

ABAB xxx

ABAB vvv relative velocity of B wrt A

ABAB vvv

ABAB aaa relative acceleration of B wrt A

ABAB aaa

2/1/2017 10:00 AM

Ball thrown vertically from 12 m

level in elevator shaft with initial

velocity of 18 m/s. At same

instant, open-platform elevator

passes 5 m level moving upward

at 2 m/s.

Determine

a. When & where ball hits the

elevator

b. Relative velocity of ball wrt

elevator at contact

Sample 11.4


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Motion of Several Particles: Dependent Motion

Position of B depends on

position of A

Rope length = constant

Sum of lengths of

segments = constant

constxx BA 2

(one DOF)


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Motion of Several Particles: Dependent Motion

constxxx CBA 22

(2 DOF)

022

022

CBA

CBA

aaa

vvv


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Sample 11.5Pulley D is attached to a collar

which is pulled down at 3 cm/s.

At t = 0, collar A starts moving

down from K with constant

acceleration and zero initial

velocity. Knowing that

velocity of collar A is 12 cm/s

as it passes L, determine the

change in elevation, velocity,

and acceleration of block B

when block A is at L.


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Assignment #11.2

33, 38, 42, 47, 52, 57


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• Given x-t curve, v-t curve = x-t curve slope

• Given v-t curve, a-t curve = v-t curve slope

Graphical Solution of

Rectilinear-Motion Problems


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Given a-t curve, change in velocity between t1 &

t2 = area under a-t curve between t1 & t2.

Given v-t curve, change in position between t1 &

t2 = area under v-t curve between t1 & t2.

Graphical Solution of

Rectilinear-Motion Problems


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Sample Problem 11.6A subway car leaves station A; it gains speed at

the rate of 4 ft/s2 for 6 s and then at the rate of 6

ft/s2 until it has reached the speed of 48 ft/s. The

car maintains the same speed until it approaches

(car does not reach B yet) station B; brakes are

then applied, giving the car a constant

deceleration and bringing it to a stop in 6 s. The

total running time from A to B is 40 s. Draw the a−t,

v−t, and x−t curves, and determine the distance

between stations A and B.


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Assignment #11.3

61, 67, 73, 79, 87


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Curvilinear Motion: Position, Velocity & Acceleration

• Curvilinear motion: Particle moving along a

curve other than a straight line

• Position vector of a particle at time t


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dt

ds

t

sv

dt

rd

t

rv

t

t

0

0

lim

lim

instantaneous velocity (vector)

instantaneous speed (scalar)


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dt

vd

t

va

t

0lim

instantaneous acceleration

(vector)

• In general, acceleration vector is

not tangent to particle path


32 2/1/2017 10:00 AM

Rectangular Components of Velocity & Acceleration

kzjyixr

kvjviv

kzjyixkdt

dzj

dt

dyi

dt

dxv

zyx

kajaia

kzjyixkdt

zdj

dt

ydi

dt

xda

zyx

2

2

2

2

2

2


2/1/2017 10:00 AM 33

Rectangular Components of Velocity & AccelerationMotion of a projectile

00 zyx agaa

initial conditions:

0000 zyx

Integrating twice:

0

02

21

00

00

zgttvytvx

vgtvvvv

yx

zyyxx

Mohammad Suliman Abuhaiba,Ph.D., P.E.2/1/2017 10:00 AM 34

Motion Relative to a Frame in Translation xyz = fixed frame of reference

Position vectors for particles A and B wrt to the

fixed frame of reference Oxyz are

: position of B wrt

moving frame Ax’y’z’ABr

ABAB rrr


35

Motion Relative to a Frame in TranslationABv

velocity of B wrt AABAB vvv

ABa

acceleration of B wrt AABAB aaa

Absolute motion of B =

combined motion of A and

relative motion of B wrt

moving reference frame

attached to A.


36

Sample Problem 11.7A projectile is fired from edge of a 150-m cliff with

an initial velocity of 180 m/s at an angle of 30°

with the horizontal. Neglecting air resistance,

find:

a. horizontal distance from the gun to the point

where the projectile strikes the ground

b. greatest elevation above the ground reached

by the projectile

2/1/2017 10:00 AM


37

Sample Problem 11.9

Automobile A is traveling

east at constant speed of

36 km/h. As automobile A

crosses the intersection,

automobile B starts from

rest 35 m north of the

intersection & moves south

with a constant

acceleration of 1.2 m/s2.

Determine position,

velocity, and acceleration

of B relative to A 5 s after A

crosses the intersection.

2/1/2017 10:00 AM


38

Assignment #11.4

89, 95, 101, 107, 113, 120, 126


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Tangential and Normal Components Velocity vector is tangent to path

ttt eee

d

ede

eee

e

tn

nnt

t

2

2sinlimlim

2sin2

00


2/1/2017 10:00 AM 40

Tangential and Normal Components

dt

ds

ds

d

d

edve

dt

dv

dt

edve

dt

dv

dt

vda t

tt

22 va

dt

dvae

ve

dt

dva ntnt

vdt

dsdsde

d

edn

t


2/1/2017 10:00 AM 41

Tangential and Normal Components Tangential component of

acceleration reflects

change of speed

Normal component reflects

change of direction

Tangential component may

be +ve or -ve

Normal component always

points toward center of

path curvatureMohammad Suliman Abuhaiba,Ph.D., P.E.

2/1/2017 10:00 AM 42

Tangential and Normal Components – 3D Path

22 va

dt

dvae

ve

dt

dva ntnt

Osculating plane: Plane

containing tangential &

normal unit vectors

ntb eee

binormale

normalprincipal e

b

n

No Acceleration component

along binormalMohammad Suliman Abuhaiba,Ph.D., P.E.

2/1/2017 10:00 AM 43

Sample 11.10A motorist is traveling on

curved section of highway at 88

m/s. The motorist applies

brakes causing a constant

deceleration rate.

Knowing that after 8 s the speed

has been reduced to 66 m/s,

determine the acceleration of

the automobile immediately

after the brakes are applied.

Mohammad Suliman Abuhaiba,Ph.D., P.E.2/1/2017 10:00 AM 44

Radial and Transverse Components

rr e

d

ede

d

ed

dt

de

dt

d

d

ed

dt

ed rr

dt

de

dt

d

d

ed

dt

edr

erer

edt

dre

dt

dr

dt

edre

dt

drer

dt

dv

r

rr

rr

rerr

Mohammad Suliman Abuhaiba,Ph.D., P.E.2/1/2017 10:00 AM

Radial and Transverse Components

errerr

dt

ed

dt

dre

dt

dr

edt

d

dt

dr

dt

ed

dt

dre

dt

rd

edt

dre

dt

dr

dt

da

r

rr

r

22

2

2

2

2


46

Radial and Transverse Components – 3D

kzeRr R

kzeReRdt

rdv R

kzeRReRR

dt

vda

R

22


2/1/2017 10:00 AM 47

Sample 11.12The rotation of the 0.9 m arm OA about

O is defined by the relation 0.15t2

where is expressed in radians and t in

seconds. Collar B slides along the arm

in such a way that its distance from O

is r = 0.9-0.12t2, where r is expressed in

meters and t in seconds. After the arm

OA has rotated through 30o , determine

a. Total velocity of collar

b. Total acceleration of collar

c. Relative acceleration of collar wrtarm


2/1/2017 10:00 AM 48

Assignment #11.5

133, 140, 146, 153, 167


49 2/1/2017 10:00 AM

Chapter 11 : Kinematics of Particles -...

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