Chapter 11 : Kinematics of Particlessite.iugaza.edu.ps/mhaiba/files/2012/01/CH-11-Kinematics... ·...

Chapter 11

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

Kinematics of Particles

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First Exam

Wednesday

2/3/2019

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

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IntroductionMechanics

Mechanics = science which

describes & predicts conditions of

rest or motion of bodies under

action of forces

Divided into three parts:

1. Mechanics of rigid bodies

2. Mechanics of deformable bodies

3. Mechanics of fluids


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Introduction

Mechanics of rigid bodies is

subdivided into:

1. Statics: bodies at rest

2. Dynamics: 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 withoutreference to cause of motion

2. Kinetics

study of relation existing between

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


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Introduction

A dynamic study could be done on twolevels:

1. Particle: object whose size & 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 & under all conditions of

loading.


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Motion of Particles

1. Rectilinear Motion

2. Curvilinear Motion


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Rectilinear Motion of Particles

Position


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9

Velocity

t

xv

t

x

t

0lim

Average velocity

Instantaneous

velocity

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

10

Instantaneous

acceleration t

va

t

0lim

t

v

Average acceleration

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11

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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13

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

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14

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

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15

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

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Sample 11.2


16

1. velocity & elevation above

ground at time t

2. highest elevation reached by

ball and corresponding time

3. time when ball will hit the

ground & corresponding

velocity

Ball tossed with 10 m/s vertical

velocity from a window 20 m

above ground. Determine:

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Sample 11.3


17

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 = - k v

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

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Assignment #11.1

1, 6, 11, 17, 22, 29


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Uniform Rectilinear Motion


19

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


20

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

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Uniformly Accelerated Rectilinear Motion


21

Acceleration of the particle is constant

020

2

020

221

2

constant

00

xxavv

xxavvdxadvvadx

dvv

x

x

v

v

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Motion of Several Particles Relative Motion


22

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

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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

1. When & where ball hits the

elevator

2. 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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A

B

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 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

• 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


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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


35

Rectangular Components of Velocity & Acceleration

Motion of a projectile

00 zyx agaa

initial conditions:

0000 zyx

Integrating twice:

0

02

21

00

00

zgttvytvx

vgtvvvv

yx

zyyxx


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 point where

projectile strikes the ground

b. greatest elevation above ground reached by

the projectile

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37

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

x’y’z’ = moving frame

position of B wrt

moving frame Ax’y’z’ABr

ABAB rrr


38

Motion Relative to a Frame in Translation

ABAB vvv

ABAB aaa

Absolute motion of B = motion of A + relative motion

of B wrt moving reference frame attached to A


Sample Problem 11.9

Car A is traveling east at

constant speed of 36 km/h.

As car A crosses the

intersection, car 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.

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40

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

ee

e

tn

nt

t

2

2sinlimlim

2sin2

00


Tangential and Normal Components

dt

ds

ds

d

d

edve

dt

dv

dt

edve

dt

dv

dt

vda

tt

t

2

2

va

dt

dva

ev

edt

dva

nt

nt

vdt

dsdsde

d

edn

t


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.

44

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.

45

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.


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

rerr



erer

edt

dre

dt

dr

dt

edre

dt

drer

dt

dv

r

rr

rr


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



49

Radial and Transverse Components – 3D

kzeRr R

kzeReRdt

rdv R

kzeRReRR

dt

vda

R

22


50

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 - 0.12 t2, where r is expressed in

meters and t in seconds. After the arm OA has rotated through

30o, determine


51

a. Total velocity of collar

b. Total acceleration of collar

c. Relative acceleration ofcollar wrt arm

Assignment #11.5

133, 140, 146, 153, 167


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Chapter 11 : Kinematics of Particlessite.iugaza.edu.ps/mhaiba/files/2012/01/CH-11-Kinematics... ·...

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