Loose Leaf For Explorations: Introduction To Astronomy
9th Edition
ISBN: 9781260432145
Author: Thomas T Arny, Stephen E Schneider Professor
Publisher: McGraw-Hill Education
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Question
Chapter 3, Problem 3TY
To determine
The contribution of Newton’s work in Kepler’s laws.
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Newton's Law of Gravitation
2. The magnitude of the acceleration of an object under the pull of Earth's gravity is
given by Newton's Universal Law of Gravitation
МЕ
a = G
R?
where G is the universal gravitational constant, ME is the mass of Earth, and R is the
distance of the object from the center of Earth.
Let x be the distance above Earth's surface. We can rewrite the formula for the
acceleration as a function of x by noting that R = Rp + x, where Rp is the radius of
Earth. Therefore,
МЕ
a(x) = G-
(RE + x)2
d.
(a) Show that
dx
1
1
(1 – x)*
- x.
(b) Use the above fact, along with the power series of
1
to determine a power
1- x
1
series for
(1+x)²*
(c) What is the radius of convergence for the series in part (b)? (Hint: You do not
need to calculate anything. What is the radius of convergence for the power series
of
1
does not change the radius of convergence.)
-? This series has the same radius of convergence since taking a derivative
Describe three propositions, now known as Kepler’s laws of planetary motion?
Gravitational Forces (A)
Gravitational, Electrical, Magnetic, and Nuclear Forces
Math Connections
Gm
The surface gravity, g, on a planet can be calculated using the formula:
g =
, where
• G = Gravity = 6.673×10-11 N • m²
kg
• m = mass in kg
• r= radius in m
Characteristics of the Planets
Planet
Mass (kg)
Radius (m)
Surface Gravity (m/s?)
Mercury
3.30 х 1023
2,440,000
3.70
Venus
4.87 x 1024
6,051,000
Earth
5.97 х 1024
9.79
Mars
6.42 x 1023
3,397,000
Jupiter
1.90 x 1027
71,492,000
Saturn
5.69 x 1026
10.45
Uranus
8.66 x 1025
25,559,000
8.84
Neptune
1.03 х 1026
24,764,000
Use the table above to answer the following questions. Insert your answers into the spaces in the
table.
9. Calculate the surface gravity on the following planets:
(6.673x10-")(4.87×10²4)_
a) Venus
g=
6,051, 000
(6.673×10"")(1.90 ×10")
71, 492, 000
b) Jupiter
g=
%3D
c) Neptune
(6.673×10-")(1.03×10%)
g=
24, 764, 000
(6.673×10 ")(6.42 ×10")
g=
d) Mars
3, 397,000
Gm
10. Calculate the radius of each of the following planets…
Chapter 3 Solutions
Loose Leaf For Explorations: Introduction To Astronomy
Ch. 3 - What is meant by inertia?Ch. 3 - (3.1) What does Newtons first law of motion tell...Ch. 3 - Explain how inertia and gravity are both involved...Ch. 3 - How does mass differ from weight?Ch. 3 - If your mass is 70 kg on Earth, what is it on the...Ch. 3 - What is Newtons law of gravity?Ch. 3 - Prob. 7QFRCh. 3 - (3.7) If you weigh 110 pounds on Earth, do you...Ch. 3 - Prob. 9QFRCh. 3 - Prob. 10QFR
Ch. 3 - Prob. 1TQCh. 3 - Prob. 2TQCh. 3 - (3.2) Is there a force of gravity between the...Ch. 3 - (3.3) Use Newtons second law of motion to explain...Ch. 3 - (3.4) How many times greater is Earths...Ch. 3 - Prob. 6TQCh. 3 - Prob. 7TQCh. 3 - Prob. 8TQCh. 3 - Prob. 9TQCh. 3 - Prob. 10TQCh. 3 - Prob. 11TQCh. 3 - (3.3) If you apply a force F to a mass m, it...Ch. 3 - Prob. 2PCh. 3 - Prob. 3PCh. 3 - Prob. 4PCh. 3 - Prob. 5PCh. 3 - (3.6) Gliese 581e is an exoplanet with a mass of...Ch. 3 - (3.7) Using the method of section 3.7, compare the...Ch. 3 - Prob. 8PCh. 3 - Prob. 9PCh. 3 - Prob. 10PCh. 3 - Prob. 11PCh. 3 - Prob. 12PCh. 3 - Prob. 13PCh. 3 - Prob. 1TYCh. 3 - Prob. 2TYCh. 3 - Prob. 3TYCh. 3 - Prob. 4TYCh. 3 - Prob. 5TYCh. 3 - Prob. 6TYCh. 3 - Prob. 7TYCh. 3 - Prob. 8TYCh. 3 - Prob. 9TY
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Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Similar questions
- A new planet is discovered orbiting a distant star. Observations have confirmed that the planet has a circular orbit with a radius of 12 AU and takes 117 days to orbit the star. Determine the mass of the star. State your answer with appropriate mks units. [NOTE: AU ..stands.for...astronomical unit". It is the average distance between Earth & the Sun. 1 AU≈ 1.496 x 1011 m.] Enter a number with units. I be quite large and your calculator will display the answer as a power of 10. If, as an example, your answer was 8.54 x 1056, you would type "8.54e56" into the answer box (remember to state your units with your answer).]arrow_forwardExplain about KEPLER’S LAWS AND THE MOTION OF PLANETS.arrow_forwardoutline kepler’s 3 laws.arrow_forward
- question about:Newton's Equation and Kepler's Law The answer has been given, please explain the process of the answer section.arrow_forwardExplain Gravitation and Kepler’s Laws of Planetary Motion?arrow_forwardExplain how Kepler was able to find a relationship (his third law) between the orbital periods and distances of the planets that did not depend on the masses of the planets or the Sun.arrow_forward
- Think back to the last time you got a new phone and had to figure out how it worked. In what way did you employ the scientific method, maybe without even realizing that you did?arrow_forwardIn what ways were the models of Ptolemy and Copernicus similar?arrow_forwardWhy did Kepler need Tycho Brahe’s data to formulate his laws?arrow_forward
- Comet Halley (Fig. P11.21) approaches the Sun to within 0.570 AU, and its orbital period is 75.6 yr. (AU is the symbol for astronomical unit, where 1 AU = 1.50 1011 m is the mean EarthSun distance.) How far from the Sun will Halleys comet travel before it starts its return journey?arrow_forwardNewton’s law of gravitation and the formula for centripetal acceleration can be used to show that: T^2=(4π^2/Gms)R^3 where G is the universal constant of gravitation and MS is the mass of the Sun. Take logarithms to base 10 of both sides of the equation to complete the expression for 2 lg T.2 lg T = ……………… × lg R + ……………………arrow_forward19. Which of the following is not one of, nor a direct consequence of, Kepler's Laws?arrow_forward
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