Chapter 6: Circular Motion and Gravitation - PowerPoint Presentation

Slide1

Chapter 6: Circular Motion and Gravitation

© 2016 Pearson Education, Inc.Slide2

Goals for Chapter 6

To understand the dynamics of circular motion.To study the unique application of circular motion as it applies to Newton's law of gravitation.

To examine the idea of weight and relate it to mass and Newton

'

s law of gravitation.To study the motion of objects in orbit as a special application of Newton's law of gravitation.

© 2016 Pearson Education, Inc.Slide3

In Section 3.4

We studied the kinematics of circular motion.Centripetal accelerationChanging velocity vector

Uniform circular motion

We acquire new terminology.

Period (T)Frequency (

f

)

© 2016 Pearson Education, Inc.Slide4

Velocity Changing from the Influence ofa

rad – Figure 6.1A review of the relationship between

and

a

rad.The velocity changes direction,not magnitude.

The magnitude of the centripetal

acceleration is:

In terms of the speed and period (time to make one complete revolution)

© 2016 Pearson Education, Inc.Slide5

Circular motion and GravitationSlide6

Details of Uniform Circular Motion

An object moves in a circle because of a centripetal net force

.

Notice how

becomes linear when Frad vanishes.

© 2016 Pearson Education, Inc.Slide7

You whirl a ball of mass

m

in a fast vertical circle on a string of length

R

. At the

top

of the circle, the tension in the string is

five

times the ball's weight. The ball's speed at this point is given by

a)

gR

b)

4

gR

c)

6

gR

d) 6 gR© 2016 Pearson Education, Inc.

Clicker questionSlide8

You whirl a ball of mass

m

in a fast vertical circle on a string of length

R

. At the

bottom

of the circle, the tension in the string is

five

times the ball's weight. The ball's speed at this point is given by

a)

gR

b)

4gR

c)

6

gR

d) 6 gR

© 2016 Pearson Education, Inc.Clicker questionSlide9

The “Giant Swing”:

Make a free body diagram of the seat including the person on it.

Find the time for one revolution for the indicated angle of 30

o

Does the angle depend on the weight of the passenger?

The person moves on a radius of

R=3+5sin30=5.5m

and





c) The net force is proportional to mass that divides out in

.The angle is independent of mass.

 Slide10
Slide11

Work out the radial acceleration of the moon around the earth.

and

g=9.8

and

So;

g

 Slide12

Ferris wheelSlide13

Top:

and

The force which the seat applies to the passenger is smaller than his weight.

For,

passenger is starting to fly off.

Bottom:

Passenger in Ferris wheel is pressed into the seat.

Ferris wheelSlide14

Model Airplane on a String – Example 6.1

How hard must you pull on the string to keep the airplane flying in a circle?T=4s m=0.5 kg

© 2016 Pearson Education, Inc.Slide15

© 2016 Pearson Education, Inc.

snowboardingSlide16

You're snowboarding down a slope. The free-body diagram in the figure represents the forces on you as you

go over the top of a mogul.go

through the bottom of a

hollow

between moguls. go along a horizontal stretch.go

along a horizontal stretch or over

the

top of

a mogul

.

© 2016 Pearson Education, Inc.

Clicker questionSlide17

Conical Pendulum Tether

Ball Problem – Example 6.2

Center-seeking Force

:

Tension

Circular MotionSlide18

Conical Pendulum

What is the period of this conical pendulum ?

,

and

So;



=

So;



(the period is independent of mass)

 Slide19

No Skidding on a Curve (I)

m

s

min

= ?

Center-seeking Force : Static FrictionSlide20

No Skidding on a Curve (II)

r

= 50.0 m

m

=

1000 kg

v

= 14.0 m/s

CSlide21

Un-Banked Curve

Given:

, m

,

and

(no skidding)

Friction force is

smaller

than radial force

(skidding)

So, no skidding at

By changing friction to

(Skidding)

The maximum safe velocity



Friction force is larger than radial forceSlide22

No Skidding on Banked Curve

F

radSlide23

No Skidding on Banked Curve

The key to this problem is to realize that the net force

causes the car to move along the curve.

and

Use;

....(1)

..........(2)

Divide equation (1) by equation (2);

Note:

For special case

=0;

and

For

special case

=

0

o

;

(unbanked curve)

 Slide24

6-54:

When

the system rotates about the rod the strings are extended as shown

. (

The tension in the upper string

is 80

N)

The block moves in a circle of radius

Each string makes an angle

with the vertical pole

This block has an acceleration of

b) What is the tension in the lower string?



c

)

What is the

speed of the block?

 Slide25

6-55 . As the bus rounds a flat curve at constant speed, a package suspended from the luggage rack on a string makes an angle with the vertical as shown.





 Slide26

Gravitation

Newton’s Law of GravitationSlide27

Gravitational attraction

Note

: Two particles of different mass exert equally strong gravitational force on each otherSlide28

Gravitational Forces (I)

‘‘Attractive

Force”

M

M

M

E

F

G

= G

r

2

M

E

M

MSlide29

Why is the Aggie not falling off the earth?

Remember there is equally strong attraction between the earth and the Aggie and vice versa

Compare the acceleration of the Aggie to the acceleration of the Earth



Forces are equal between the Aggie and the Earth

=g



with

(Aggie’s mass)

=

g

 Slide30

Cavendish balance

Cavendish(1798) announced that he has weighted the earthSlide31

Cavendish Tension balance (1798)

Air current in the room is negligible to the gravitational attraction force

(Torsion force)

and

When torsion and gravitational forces are in equilibrium;



Molecular motors (kinetics);

 Slide32

Gravitational Force Falls off Quickly – Figure 6.15

The gravitational force is proportional to 1/

r

2

, and thus the weight of an object decreases inversely with the square of the distance from the earth's center (not distance from the surface of the earth

).

© 2016 Pearson Education, Inc.Slide33

Earth mass

and radius



When radius is variable like

with variable mass

of Earth.

Then;

and

What is the magnitude of the gravitational force inside, on the surface, and outside the earth??Slide34

g

= 9.80 m/s2R

E

= 6.37

x 10

6

m

M

E

= 5.96

x

1024

kg

rE = 5.50

x 10

3

kg/m

3

= 5.50 g/cm3 ~ 2

x rRock

Average Density of the EarthSlide35

Satellite Motion: What Happens When Velocity Rises?

Eventually, F

g

balances and you have orbit.

When is large enough, you achieve escape velocity.An orbit is not fundamentally different from familiar trajectories on earth. If

you launch it slowly, it falls back

. If

you launch it fast enough, the earth curves away from it as it falls, and it goes into orbit

.

© 2016 Pearson Education, Inc.Slide36

Circular Satellite Orbit Velocity

If a satellite is in a perfect circular orbit with speed orbit, the gravitational force provides the centripetal force needed to keep it moving in a circular path.

© 2016 Pearson Education, Inc.Slide37

Circular orbit period

The larger r then slower the speed and the larger the periodSlide38

Example 6.10

:

Earth mass

and radius

a) What is the speed?



b) When is the period?

c) What is the radial acceleration?

Weather

SatelliteSlide39

Height above the surface of Earth.

Earth mass

and radius



and



=36000 km

b) What is the velocity?



Geo-synchronous Satellite (at the equator of Earth)

Not to scaleSlide40

If an Object is Massive, Even Photons Cannot Escape

A "black hole

"

is a collapsed sun of immense density such that a tiny radius contains all the former mass of a star.

The radius to prevent light from escaping is termed the "Schwarzschild Radius

."

The edge of this radius has even entered pop culture in films. This radius for light is called the

"

event horizon

."

© 2016 Pearson Education, Inc.Slide41

Hawking @ HS ranch

© 2016 Pearson Education, Inc.Slide42

Black hole

Steven Hawkins is associated with the department of Physics and Astronomy at TAMU

Sun mass

and radius

Average density of Sun;

40% denser than water

Temperature: 5800

o

K at surface and (1.5x10

7

)

o

K in the interior of Sun. (highly ionize plasma gas)

 Slide43

Escape velocity from the Sun

(mass

and radius

)



and

How can the sun become a Black Hole?

If the radius of the sun become 500 times larger for the same density, then the light could not escape. This would increase the sun’s mass. In another word sun became Black Hole.

There is a second way, which is to decrease the radius of sun.

c

(Schwarzschild radius)

For

 light can be emitted

For



no light

can be

emitted

(Black hole)

 Slide44

To what fraction of sun’s current radius would the sun have to be compressed to become a black hole?



 Slide45

operates

two gravitational wave observatories in unison: the LIGO Livingston Observatory

in

Livingston, Louisiana

, and the LIGO Hanford Observatory, on the

DOE Hanford Site

,located

near

Richland, Washington

. These sites are separated by 3,002 kilometers (1,865 miles)

Laser Interferometer Gravitational-Wave Observatory

(

LIGO

)

Collison of two black holes 1.3 billion years ago, each black hole was about 30 times mass of the Sun, and 3 solar mass were converted to gravitational waves.