P1710_MWF07

doutfanaticalMechanics

Nov 14, 2013 (3 years and 7 months ago)

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Free Fall:



Kinematics in two (or more) dimensions obeys the
same 1
-

D equations in each component
independently.

Physics 1710

Chapter 4: 2
-
D Motion

II

a
x

= 0

a
y

=
-
g

v
x

=constant

y = y
initial


½ gt
2

x = x
initial
+ v
x,initial

t

Kinematic Equations

Physics 1710

Chapter 4: 2
-
D Motion

II

REVIEW

x(t) = x
initial
+ v
initial
t + 1/2 a
o

t
2

v(t) = dx/dt = v
initial

+ a
o

t

a= dv/dt = a
o


Observe Air Track

1


Lecture:




The velocity and acceleration of a body in a moving
(and accelerating) frame of reference (FoR ) is equal to
that of a stationary FoR
minus

the velocity or acceleration
of the moving FoR.



v


=

v

=
v
frame of reference



a


=

a

=
a
frame of reference




Motion in a circle at a constant speed is due to an
acceleration toward the center of the circle, a
centripetal

acceleration of



a =
-

ω
2

r
and
|
a
|
= v
2
/
|r|

toward the center.


Physics 1710

Chapter 4: 2
-
D Motion

II

Frame of Reference




Physics 1710

Chapter 4: 2
-
D Motion

II

Physics 1710

Chapter 4: 2
-
D Motion

II

Fly

v


v


v
frame of reference

v

v = v
′ + v
frame of reference


and

v

= v

-

v
frame of reference


Relative Motion



and the Galilean Transformation:





r

=
㴠=
r
=


v
frame of reference

t


d
r

⽤/=㴠==
r
⽤/=


v
frame of reference



v


=

v

=
v
frame of reference




d
v

⽤/
=
=

=

v
⽤/
=

=

v
frame of reference
/dt



a

=

=


=
a
frame of reference


Physics 1710

Chapter 4: 2
-
D Motion

II

Relative Motion


in a Free Falling Frame of Reference

Physics 1710

Chapter 4: 2
-
D Motion

II

In Lab
Frame

In Moving
Frame

a

=

=


=
a
frame of reference


Everyday Physics:




When a bird is flying at the same
velocity(same speed and direction) as a car
what is its relative velocity

v

to=the=car?
=


When you brake an automobile, which
direction is the acceleration on the vehicle?
Which direction do the passengers sense as
the acceleration relative to the frame of
reference of the car?

Physics 1710

Chapter 4: 2
-
D Motion

II

Relative Motion



and the Galilean Transformation:







v


=

v

=
v
frame of reference





a


=

a

=
a
frame of reference


Physics 1710

Chapter 4: 2
-
D Motion

II

Uniform Circular Motion




θ

r

Physics 1710

Chapter 4: 2
-
D Motion

II

x = R sin
θ

y = R cos θ

R
2

= x
2

+ y
2

r
=
(x,y) = x
i

+ y
j

v

v
x


= R cos
θ d θ/dt


= Rω cos θ

v
y


=
-

R sin
θ d θ/dt


=
-

Rω sin θ

d θ/dt = ω = constant

Uniform Circular Motion




θ

r

Physics 1710

Chapter 4: 2
-
D Motion

II

v

v
x


= R cos
θ d θ/dt


= R ω cos θ

v
y


=
-

R sin
θ d θ/dt


=
-

R ω sin θ





a
x


=
-

R
ω
2

sin
θ

a
y


=
-

R
ω
2

cos
θ

a

=
-

ω
2

r


|
a
|

= v
2
/R, toward center

Uniform Circular Motion

in review



a
?ÿ

= a,

a constant value always pointing toward the center of the
circle:
Centripetal acceleration
.

a

= a
x

i
+ a
y
j =
-

ω
2
r

where

a
x

= a sin
θ =
-

(
v
2
/R) sin
θ =
-

ω
2

R
sin
θ

a
y

= a cos
θ =
-

(
v
2
/R) cos
θ =
-

ω
2

R
cos
θ


Physics 1710

Chapter 4: 2
-
D Motion

II

Uniform Circular Motion


|
a
|

= v
2

/R, toward center


The
Centripetal
acceleration, where v is the tangential
speed and R is the radius of the circle.

v =
ω R =
2
π
R / T ,


Where T is the “
period
” or time to make one revolution.

|
a

|
= 4
π
=
2
R/ T

2

=
ω
2

R


Physics 1710

Chapter 4: 2
-
D Motion

II

Uniform Circular Motion

Little Johnny on the Farm

part II




.

Physics 1710

Chapter 4: 2
-
D Motion

II

-

g

a

v

-

g

a

v

.

?

Uniform Circular Motion




-

g

a
FoR

Physics 1710

Chapter 4: 2
-
D Motion

II

In Frame of
Reference of Bucket

a
′ =
-
g


a
FoR




If

|a
FoR
| ≽ |g|

In same (down)
direction,

a
′ is up!

Uniform Circular Motion




g

a

Physics 1710

Chapter 4: 2
-
D Motion

II

a = (2
π/T)
2

R,

T

1. sec

R

1. m

2
π

6.

a


(6./1.sec)
2

(1.m)

a
≈ 36. m/sec
2

a
> 3 g


Do you believe
this?

80/20 Summary:




In a moving or accelerating Frame of Reference



v

=

=


=
v
frame of reference



a
?¿?

=

a

=
a
frame of reference




The
Centripetal

acceleration is



a =
-


2

r




or
|
a
|
= v
2
/
|r|,

toward the center.




Physics 1710

Chapter 4: 2
-
D Motion

II