equations of motion for a particle

8/8/2019 Equations of Motion for a Particle

1/17

Rectilinear Motion: Variable acceleration

If a mathematical relation is established between any two of the four

variables s, v, a, and t, then a third variable can be determined by using one

of the following equations which relates all three variables.

adsvdva

dv

v

dsdt

dt

dva

dt

dsv

!p!!

!

!

IfAcceleration is constant:

2

0 210 tatvss c! tavv c! 0 )(2 0

2

0

2 ssavv c !

Review of Kinematics


2/17

Cartesian Rectangular x, y z coordinates:

These coordinates are often used when the motion can be resolved into

rectangular components. They are also very useful for studying projectilemotion since the acceleration of the projectile is always downward

za

ya

xa

z

y

x

!

!

!

zv

yv

xv

z

y

x

!

!

!


3/17

n, t, b Intrinsic coordinates

These coordinates are particularly advantageous for studying

particle acceleration along a known path. This is because the t andn components of a represent the separate changes in magnitude

and direction of the velocity, respectively, and these components

can be readily formulated

V2va

dt

dva

dtdsv

n

t

!

!

!

vdvdsat !

-

-

!

2

2

2/32

1

dx

yd

dx

dy

V)(xfy !When is known


4/17

zr ,,U Coordinates

These coordinates are used when the data regarding the angular motion ofthe radial coordinate r is given to describe the particles motion. Also, some

paths of the motion can conveniently be described using these coordinates.

zv

rv

rv

z

r

!

!

!

UU

za

rra

rra

z

r

!

!!

UU

U

U2

2


5/17

!

!

!

zz

yy

xx

maF

maF

maF

!

!

!

0b

nn

tt

a

a

!

!

!

zz

rr

maF

maF

maF

UU


6/17

!

!

!

zz

yy

xx

a

a

a

Equations of Motion of a particle in RectangularCartesian Coordinate System

za

ya

xa

z

y

x

!

!!


7/17

!!

!

0b

nn

tt

a

a

V

2va

dt

dv

a

dt

dsv

n

t

!

!

! vdvdsat !

Equations of Motion of a particle in Intrinsic Coordinate Systems


8/17

!

!

!

zz

rr

maF

maF

maF

UU

zv

rv

rv

z

r

!

!

!

UU

za

rra

rra

z

r

!

!

!

UU

U

U2

2

Equations of Motion of a particle in Cylindrical and PolarCoordinate Systems

In polar coordinate:

0

0

!

!

z

z

a

v


9/17

Geometric requirements Kinematics (position of the particle s as

function of the spatial variable, velocity, and acceleration)

adsvdva

dv

v

dsdt

dt

dva

dt

dsv

!p!!

!

!

Dynamic Requirements

mvp

madt

dvm

dt

mvd

dt

dpf

!

!!!!)(

Nf

mgf

ksf

mvp

kf

g

s

Q!

!

!

!

Constitutive requirements for system elements and gravitational field

p is linear momentum

Momentum - velocity

Spring force

Gravitational force

Frictional force

Example: Rectilinear Motion with Variable acceleration

Requirements for Solving Dynamic Problems


10/17

Example1: Consider a simple pendulum consisting of a mass, m, suspended from a

string of length L and negligible mass. We can formulate the problem in polar

coordinates, and noting that r = L (constant), and write r and theta components.

T

mg

U

ru

Uu

x

y

L

o

UUU

U

rra

rrar

2

2

!!

0!!! rrLr

UU

UU

U

mLmgF

mLTmgFr

!!

!!

sin

cos2

Where T is the tension on the string. If we restrict the

motion to small motion, we can approximate,

UU }sin

The equationU becomes 0! UUL

g

max)0( UU !

tL

gCt

L

gCt

! sincos)( 21UIntegrating we obtain the general solution

Where the constants C1 and C2 are determined from the initial conditions. Thus, if

t

L

gt

! cos)(

maxUU 2cos UU mLmgT !and


11/17

The ball has a mass of 30 kg and speed v = 4 m/s at the instant it

is at its lowest point, Determine the tension in the cord and

the rate at which the balls speed is decreasing at the instant Q20!U

Q

0!U

Neglect the size of the ball

Problem 13-62:


12/17

Given v0=4 m/s,

determine v and tension T

n

tT

tunu

Umg

ttt

nn

amaF

vTmaF

30sin)81.9(30

430cos)81.9(30

2

!!

!!

U

U

V

2va

dt

dva

dt

dsv

n

t

!

!

!

4!!l

V

vdvdsat !

We can formulate the problem

using n-t coordinates:

vdvdsat ! UUU ddsrs 44 !p!!

!!vv

v

vdvvdvd400

)4(sin81.9 UUU

)1(cos4.3982

14

2

1

2

1cos)4(81.9

222

0!! UU U vv

NT

sma

smv

t

361

/36.3

/357.3

2

!

!

!

At Q20!U


13/17


14/17


15/17

s

1


16/17


17/17

equations of motion for a particle

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