Chapter 11 : Kinematics of Particlessite.iugaza.edu.ps/.../CH-11-Kinematics-of-Particles-22.pdf ·...

Chapter 11

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

Kinematics of Particles

2/20/2015 11:21 AM

Introduction Mechanics

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 fluids

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

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Introduction

Mechanics of rigid bodies is subdivided 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 reference to the cause of motion

2. Kinetics

study of the relation existing between the

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


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Introduction

A dynamic study could be done on two 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.


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Motion of Particles

Motion of Particles:

1. Rectilinear Motion

2. Curvilinear Motion


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Rectilinear Motion of Particles

Position


7

Velocity

t

xv

t

x

t

0lim

Average velocity

Instantaneous

velocity

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Acceleration


8

Instantaneous

acceleration t

va

t

0lim

t

v

Average acceleration

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9

• Consider particle with motion given by

326 ttx 2312 ttdt

dxv

tdt

xd

dt

dva 612

2

2

• at t = 0, x = 0, v = 0, a = 12 m/s2

• at t = 2 s, x = 16 m, v = vmax = 12 m/s, a = 0

• at t = 4 s, x = xmax = 32 m, v = 0, a = -12 m/s2

• at t = 6 s, x = 0, v = -36 m/s, a = 24 m/s2

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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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Determination of the Motion of a Particle


11

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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12

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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13

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


14

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.

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Sample 11.3


15

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; that is a = -kv

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

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Assignment #11.1

1, 6, 11, 17, 22, 29 Due Wednesday

11/2/2015


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Uniform Rectilinear Motion


17

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


18

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


19

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

a. when and where ball hits the

elevator

b. relative velocity of ball wrt

levator at contact

Sample 11.4


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Motion of Several Particles: Dependent Motion

Position of block B depends on

position of block A.

Since rope is of constant length, it

follows that sum of lengths of

segments must be constant.

BA xx 2 constant (one DOF)


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Motion of Several Particles: Dependent Motion

CBA xxx 22 constant (2 DOF)

022or022

022or022

CBACBA

CBACBA

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


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Sample 11.5 Pulley 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

Due Saturday 14/2/2015


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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 11.6 A 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 Due Monday 16/2/2015


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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 &

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

31 2/20/2015 11:21 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


2/20/2015 11:21 AM 32

Rectangular Components of Velocity & Acceleration

Motion of a projectile

00 zagyaxa zyx

initial conditions:

0000 zyx

Integrating twice:

0

02

21

00

00

zgttvytvx

vgtvvvv

yx

zyyxx

Motion in horizontal direction is uniform

Motion in vertical direction is uniformly accelerated


2/20/2015 11:21 AM 33

Motion Relative to a Frame in Translation

xyz = fixed frame of reference

moving frames of reference: frames not rigidly

attached to the fixed reference frame

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


. and BA rr

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Motion Relative to a Frame in Translation

ABv

velocity of B wrt A ABAB vvv

ABa

acceleration of B wrt A ABAB aaa

Absolute motion of B =

combined motion of A and

relative motion of B wrt

moving reference frame

attached to A.


2/20/2015 11:21 AM 35

Sample Problem 11.7 A 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.

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36

Sample Problem 11.9

Automobile A is traveling east

at the constant speed of 36

km/h. As automobile A crosses

the intersection shown,

automobile B starts from rest 35

m north of the intersection and

moves south with a constant

acceleration of 1.2 m/s2.

Determine the position,

velocity, and acceleration of B

relative to A 5 s after A crosses

the intersection.

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37

Assignment #11.4

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

Due Wednesday 18/2/2015


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Tangential and Normal Components Velocity vector is tangent to path.

= tangential unit

vectors for particle path at P &

P’

tt ee and

ttt eee

d

ede

eee

e

tn

nnt

t

2

2sinlimlim

2sin2

00


2/20/2015 11:21 AM 39

Tangential and Normal Components

dt

ds

ds

d

d

edve

dt

dv

dt

edve

dt

dv

dt

vda tt

22 va

dt

dvae

ve

dt

dva ntnt

vdt

dsdsde

d

edn

t


2/20/2015 11:21 AM 40

Tangential and Normal Components Tangential component of acceleration reflects

change of speed

Normal component reflects change of direction.

Tangential component may be positive or negative.

Normal component always points toward center of

path curvature.


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Tangential and Normal Components

22 va

dt

dvae

ve

dt

dva ntnt

Relations for tangential & normal acceleration

also apply for particle moving along space curve.

Osculating plane: Plane containing

tangential & normal unit vectors

ntb eee

binormale

normalprincipal e

b

n

No Acceleration component along

binormal. Mohammad Suliman Abuhaiba,Ph.D., P.E. 2/20/2015 11:21 AM 42

Sample 11.10

A 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. 2/20/2015 11:21 AM 43

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

The particle velocity vector is

rerr

Mohammad Suliman Abuhaiba,Ph.D., P.E. 2/20/2015 11:21 AM

Radial and Transverse Components

The particle acceleration vector is

errerr

dt

ed

dt

dre

dt

dre

dt

d

dt

dr

dt

ed

dt

dre

dt

rd

edt

dre

dt

dr

dt

da

r

rr

r

22

2

2

2

2


2/20/2015 11:21 AM 45

Radial and Transverse Components – 3D

kzeRr R

kzeReRdt

rdv R

kzeRReRR

dt

vda

R

22


2/20/2015 11:21 AM 46

Sample 11.12 The 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 the collar

b. total acceleration of the collar

c. relative acceleration of the collar wrt the arm

Mohammad Suliman Abuhaiba,Ph.D., P.E. 2/20/2015 11:21 AM 47

Assignment #11.5

133, 140, 146, 153, 167

Due Saturday 21/2/2015

First Exam: Monday

23/2/2015


48 2/20/2015 11:21 AM

Chapter 11 : Kinematics of Particlessite.iugaza.edu.ps/.../CH-11-Kinematics-of-Particles-22.pdf ·...

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