#### Transcript Chapter 7: Circular Motion and Gravitation

Section 3: Motion in Space

### Objectives

Describe

motion.

Kepler’s laws of planetary 

Relate

Newton’s mathematical analysis of gravitational force to the elliptical planetary orbits proposed by Kepler.

Solve

problems involving orbital speed and period.

### Kepler’s Laws

Kepler’s laws

planets.

describe the motion of the   

First Law:

Each planet travels in an elliptical orbit around the sun, and the sun is at one of the focal points.

Second Law:

An imaginary line drawn from the sun to any planet sweeps out equal areas in equal time intervals.

Third Law:

The square of a planet’s orbital period (

T

2 ) is proportional to the cube of the average distance (

r

3 ) between the planet and the sun.

### continued

Kepler’s laws

were developed a generation before Newton’s law of universal gravitation.

Newton

demonstrated that Kepler’s laws are consistent with the

law of universal gravitation

.

 The fact that Kepler’s laws closely matched observations gave additional support for Newton’s theory of gravitation.

### continued

According to

Kepler’s second law,

if the time a planet takes to travel the arc on the left (∆

t 1

) is equal to the time the planet takes to cover the arc on the right (∆

t 2

), then the area

A 1

is equal to the area

A 2 .

Thus, the planet travels faster when it is closer to the sun and slower when it is farther away.

### continued

Kepler’s third law

states that

T

2 

r

3 .

 The constant of proportionality is 4 p 2 /

Gm

, where

m

is the mass of the object being orbited.

 So, Kepler’s third law can also be stated as follows:

T

2     4 p 2

Gm

 

r

3

### continued

Kepler’s third law

leads to an equation for the

period

of an object in a circular orbit. The

speed

of an object in a circular orbit depends on the same factors:

T

 2 p

r

3

Gm v t

G m r

• Note that

m

is the mass of the central object that is being orbited.

The mass of the planet or satellite that is in orbit does not affect its speed or period.

• The mean radius (

r

) is the distance between the centers of the two bodies.

### Sample Problem

Period and Speed of an Orbiting Object

Magellan was the first planetary spacecraft to be launched from a space shuttle. During the spacecraft’s fifth orbit around Venus, Magellan traveled at a mean altitude of 361km. If the orbit had been circular, what would Magellan’s period and speed have been?

### continued

1. Define

Given:

r 1

= 361 km = 3.61 Unknown:

T

= ?

v t

= ?

 10 5 m

2. Plan Choose an equation or situation:

equations for the period and speed of an object in a circular orbit.

Use the

T

 2 p

r

3

Gm v t

Gm r

### continued

Use

Table 1

in the textbook to find the values for the radius (

r2

) and mass (

m

) of Venus.

r2

= 6.05  10 6 m

m =

4.87  10 24 kg Find

r

by adding the distance between the spacecraft and Venus’s surface (

r1

) to Venus’s radius (

r2

).

r

=

r1 + r2 r =

3.61  10 5 m + 6.05  10 6 m = 6.41  10 6 m

### continued

3. Calculate

T

 2 p

r

3

Gm

=2 p

T

 5.66

 10 3 s (6.41

 10 6 m) 3 (6.673

 10 –11 N•m 2 /kg 2 )(4.87

 10 24 kg)

v t v t

 

Gm r

 (6.673

 10 –11 N•m 2 /kg 2 )(4.87

 10 24 kg) 6.41

 10 6 m 7.12

 10 3 m/s

4. Evaluate

Magellan takes (5.66  one orbit.

10 3 s)(1 min/60 s)  94 min to complete

### Weight and Weightlessness

To learn about apparent weightlessness, imagine that you are in an elevator:

 When the elevator is at rest, the magnitude of the normal force acting on you equals your weight.  If the elevator were to accelerate downward at 9.81 m/s 2 , you and the elevator would both be in free fall. You have the same weight, but there is no normal force acting on you.

 This situation is called

apparent weightlessness.

 Astronauts in orbit experience apparent weightlessness.