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Department

Basic Engineering

Course Code

BE 1101

Professor

All

Chapter

Permachart

MECHANICS

Physics

Physics

permacharts.com

VELOCITY & ACCELERATION

PARTICLE DYNAMICS

Y

X

s

a

v

θ

v

x

v

y

θ

s

r

aR

aT

v

vvv

vdx

dt v

vdy

dt v

xy

x

y

=+ ==

==

22cos

sin

θ

θ

vds

dt rd

dt

==

θ

adv

dt

adv

dt a

adv

dt a

xx

y

y

===

==

cos

sin

θ

θ

advdtg

vdv

dt dt gt h vdt gt

tt

===

====

∫∫

981

1

2

00

2

.m/s

2

Angular velocity

Note:

a= Constant (e.g., free fall)

Acceleration Velocity Diagrams

Angular

acceleration

Tangential

acceleration

Centripetal

acceleration

a = Constant

αω

=ddt

arddt

T=()

ω

av

r

R=2

ωω α

θω α

=+

=+

t

tt

1

2

2

1573955 rad/s RPM=°=./.s

ωθ

==d dt v r

Rectilinear Motion Circular Motion

ENERGY

Work/Power/Kinetic Energy

Rectilinear Motion Circular Motion

Equilibrium Momentum & Force Inertia

Rectilinear Motion Circular Motion

xxmmxdmdm

yymmydmdm

cm i i i

cm i i i

..

..

=⇒

∫∫

=⇒

∫∫

∑∑

∑∑

Center of Mass (Gravity)

Force (Newton’s Second Law)

Imr rdm

ImR ImR

ii

=⇒

∫

==

∑22

22

2

5

1

2

Sphere: Cylinder:

Moment of Inertia

Torque

Gravitational Force Centripetal Force

Transitional Equilibrium Rotational Equilibrium

Conservation of linear

momentum

Conservation of angular

momentum

Linear Momentum Angular Momentum

pmv=

LI=

ω

τ

ω

==

==

••

Fr

dL

dt Id

dt

Force Lever Arm

Fmv

rmr

C==

22

ω

FG

mm

rG

G==•−

12

2

11

66710.Nm

kg

2

2

Fmv

rmr

C==

22

ω

FFandF

xy

∑∑ ∑

== =000( )

τ

∑==011 2 2

()Lever Rule:Fr F r

p

F

system

external

∑

∑

=…

=

constant

if 0

L

system

external

∑

∑

=…

=

constant

if

τ

0

WFs=cos

θ

Wmgh=(Potential Energy)

Kmv=1

2

2

PdW

dt Fv= =

WI

d

dt

==

τθ ωθ

PdW

dt

d

dt Id

dt

== =

τθωω

Work (Constant Force)

Work (gravity)

Power (Constant Force)

Kinetic energy

Power (Constant Torque)

Work (Constant Torque)

HARMONIC MOTION

Note: Conservation of

mechanical energy

ﬁ∑K+ ∑U= constant

KI=1

2

2

ω

Kinetic energy

xA t= +sin( )

ωφ

vA t= +

ωωφ

cos( )

aAt=−+

ωωφ

2sin( )

Tv

m

k

== =

12 2

π

ωπ

m

m

m

x = A v = 0 a = – A

2

x = 0 v = A a = 0

x = –A v = 0 a = A

2

ω

ω

ω

Total energy

constant

=

=+=

1

2

2

kA

UK

FkxUFdxkx

Kmv

=− = =

∫

=

1

2

2

1

2

2

ωπν

=2

A= amplitude

f

= phase

angle

U= potential

energy

n

= frequency

K= kinetic

energy

k= spring

constant

m= mass

Displacement

Velocity

Mass On Spring

Acceleration

Period

Angular frequency

FLUID MOTION: CONSTANT RATE & DENSITY

h1

h

A

v

p

2

2

2

2

A

v

p

1

1

1

QvA vA==⇒ =

11 2 2 Flow In Flow Out

pgh vp gh v

pgh v

11

1

21

222

1

22

2

1

2

2

++ =+ +

⇒+ + =

ρρ ρ ρ

ρρ

constant

vgh

22≅[()]

∆

vv p gh

12

0===

∆∆ρ

Q= flow rate

r

= density

p= pressure = force/area

Continuity Equation

Bernoulli Equation

Torricelli’s Equation

(v1≈ 0)

Hydrostatic Pressure

MAGNETISM

Bi

d

d

=

µ

π

2

(at )

BiNl=

µ

(at axis)

ε

==

W

qBlv

ε

=−Nd AB

dt

()

ε

=−Ldi

dt

System Magnetic Field Produced

Straight wire of length l,

carrying current, i

Coil Nturns, length l,

carrying current i

System EMF Induced

Wire moving in magnetic field

Coil in time varying

magnetic field

Self-induction

L= inductance

(henry, H): 1 H =

1 volt-second/ampere

B = magnetic field

(tesla, T): 1 T = 104

gauss = 1 newton/

ampere-meter =

1 weber/m2:

m°= permeability

MAGNETIC FIELDS PRODUCED BY CURRENTS CURRENTS INDUCED BY MAGNETIC FIELDS

Bernoulli Equation

l e a r n • r e f e r e n c e • r e v i e w • i n s t a n t l y

TM

permacharts

PHYSICS • A-733-11© 2002-2012 Mindsource Technologies Inc.

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