Normal & Tangential (n-t) Coordinates, Schemes and Mind Maps of Dynamics

Download Normal & Tangential (n-t) Coordinates and more Schemes and Mind Maps Dynamics in PDF only on Docsity!

Normal & Tangential (

n-t

)

Coordinates^ Lecture 4ME 231: Dynamics

Question of the Day^ A particle moves in acircular path of radius^ r^ = 0.8 m with constant speed( v ) of 2 m/s. The

velocity

undergoes a vector change

 v

from

A^ to

B.

ME 231: Dynamics

Express the magnitude of

 v^

in terms of

v^ and

. Express the time interval

 t^

in terms of

v ,

, and

r. Obtain the magnitude of average acceleration by computing

 v /

 t.

Recall: Possible Coordinate Systems^  Rectangular (

x ,^ y

,^ z )

 Polar (

r ,^ 

,^ z )

 Spherical (

R ,^ 

,^ )

 Normal andTangential (

n ,^ t

) ME 231: Dynamics

^ Rectangular (

x ,^ y

,^ z )

 Polar (

r ,^ 

,^ z )

 Spherical (

R ,^ 

,^ )^ 

N-T Vector Representation^  The

n - and

t -coordinates move along the path

with the particle Tangential

coordinate is parallel to the

velocity  The positive direction for the

normal

coordinate is toward the center of curvature

ME 231: Dynamics

Path variables along the tangent (

t ) and normal (

n )

? t ^ e

a ? a

Acceleration

ME 231: Dynamics

^ d

ds^ 

t t t^

e e e v

^

v ds dt

t t t^

e e e v a^

v v vd dt d^ dt

     n n t

t^

e e

e e^

^

^

d dt d^ dt

t n^

e e a

2

v v^

v e^ n 

v^ ^

v^ e^ t

Velocity and Acceleration: Exercise

ME 231: Dynamics

A car passes through a dip in the road at

A^ with

constant

speed

( v ) giving it an

acceleration

( a )

equal to 0.

g. The

radius of curvature

(^ ) at

A

is 100 m and the distance from the road to themass center

G^ of the car is 0.6 m.

Determine the

speed

( v ) of the car.

t n^

e e a

2

v v^

Geometric Interpretation^ ^

Normal

component is always directedtoward center ofcurvature Tangent

component is directed toward+ t^ (or -

t ) directionif speed isincreasing(or decreasing)

ME 231: Dynamics

t n^

e e a

2

v v^

Circular Motion^  Radius of curvature



becomes constant

r

 Angle

is replaced by angle



ME 231: Dynamics

an r O v at P t n

 r

v^ 

^

^

v r v r a^ n^

^

2 2 

^ rv at^

v a  n^

2 v 

a  t^

Ý v^ 

ÝÝ^ 



Outline for Today^  Question of the day N-T vector representation Velocity and acceleration Geometric interpretation Circular motion Answer your questions!

ME 231: Dynamics

^ Question of the day N-T vector representation Velocity and acceleration Geometric interpretation Circular motion

For Next Time…^  Begin Homework #2 due next week

(9/5) Read Chapter 2, Section 2.

ME 231: Dynamics