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.