#### 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.

### Kepler’s Laws,

*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.

### Kepler’s Laws,

*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.*

### Kepler’s Laws,

*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

### Kepler’s Laws,

*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.

### Planetary Data

### 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?*

### Sample Problem,

*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*

### Sample Problem,

*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

### Sample Problem,

*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.