Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is M = ( 4 π 2 G ) ( a 3 T 2 ) Proof: Using Kepler’s Third Law, we can state that T 2 ∝ a 3 . A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is: ⇒ T 2 = ( 4 π 2 G ( M + m ) ) a 3 The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as: ⇒ T 2 = ( 4 π 2 G ( M ) ) a 3 ⇒ T 2 = ( 4 π 2 G M ) a 3 ⇒ M = ( 4 π 2 G ) ( a 3 T 2 )

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Answer 1

Question

Chapter 14.6, Problem 2E

To determine

To show:

Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is $M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}})$

Proof:

Using Kepler’s Third Law, we can state that $T^{2} \propto a^{3}$ .

A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is:

$\Rightarrow T^{2} = (\frac{4 π^{2}}{G (M + m)}) a^{3}$

The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as:

$\begin{array}{l} \Rightarrow T^{2} = (\frac{4 π^{2}}{G (M)}) a^{3} \\ \Rightarrow T^{2} = (\frac{4 π^{2}}{G M}) a^{3} \\ \Rightarrow M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}}) \end{array}$

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Answer 2

Question

Chapter 14.6, Problem 2E

To determine

To show:

Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is $M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}})$

Proof:

Using Kepler’s Third Law, we can state that $T^{2} \propto a^{3}$ .

A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is:

$\Rightarrow T^{2} = (\frac{4 π^{2}}{G (M + m)}) a^{3}$

The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as:

$\begin{array}{l} \Rightarrow T^{2} = (\frac{4 π^{2}}{G (M)}) a^{3} \\ \Rightarrow T^{2} = (\frac{4 π^{2}}{G M}) a^{3} \\ \Rightarrow M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}}) \end{array}$

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Answer 3

Question

Chapter 14.6, Problem 2E

To determine

To show:

Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is $M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}})$

Proof:

Using Kepler’s Third Law, we can state that $T^{2} \propto a^{3}$ .

A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is:

$\Rightarrow T^{2} = (\frac{4 π^{2}}{G (M + m)}) a^{3}$

The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as:

$\begin{array}{l} \Rightarrow T^{2} = (\frac{4 π^{2}}{G (M)}) a^{3} \\ \Rightarrow T^{2} = (\frac{4 π^{2}}{G M}) a^{3} \\ \Rightarrow M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}}) \end{array}$

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Answer 4

Question

Chapter 14.6, Problem 2E

To determine

To show:

Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is $M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}})$

Proof:

Using Kepler’s Third Law, we can state that $T^{2} \propto a^{3}$ .

A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is:

$\Rightarrow T^{2} = (\frac{4 π^{2}}{G (M + m)}) a^{3}$

The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as:

$\begin{array}{l} \Rightarrow T^{2} = (\frac{4 π^{2}}{G (M)}) a^{3} \\ \Rightarrow T^{2} = (\frac{4 π^{2}}{G M}) a^{3} \\ \Rightarrow M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}}) \end{array}$

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Answer 5

Chapter 14.6, Problem 2E

To determine

To show:

Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is $M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}})$

Proof:

Using Kepler’s Third Law, we can state that $T^{2} \propto a^{3}$ .

A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is:

$\Rightarrow T^{2} = (\frac{4 π^{2}}{G (M + m)}) a^{3}$

The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as:

$\begin{array}{l} \Rightarrow T^{2} = (\frac{4 π^{2}}{G (M)}) a^{3} \\ \Rightarrow T^{2} = (\frac{4 π^{2}}{G M}) a^{3} \\ \Rightarrow M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}}) \end{array}$

Answer 6

Chapter 14.6, Problem 2E

To determine

To show:

Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is $M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}})$

Proof:

Using Kepler’s Third Law, we can state that $T^{2} \propto a^{3}$ .

A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is:

$\Rightarrow T^{2} = (\frac{4 π^{2}}{G (M + m)}) a^{3}$

The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as:

$\begin{array}{l} \Rightarrow T^{2} = (\frac{4 π^{2}}{G (M)}) a^{3} \\ \Rightarrow T^{2} = (\frac{4 π^{2}}{G M}) a^{3} \\ \Rightarrow M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}}) \end{array}$

Answer 7

Chapter 14.6, Problem 2E

To determine

To show:

Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is $M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}})$

Proof:

Using Kepler’s Third Law, we can state that $T^{2} \propto a^{3}$ .

A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is:

$\Rightarrow T^{2} = (\frac{4 π^{2}}{G (M + m)}) a^{3}$

The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as:

$\begin{array}{l} \Rightarrow T^{2} = (\frac{4 π^{2}}{G (M)}) a^{3} \\ \Rightarrow T^{2} = (\frac{4 π^{2}}{G M}) a^{3} \\ \Rightarrow M = (\frac{4 π^{2}}{G}) (\frac{a^{3}}{T^{2}}) \end{array}$

Answer 8

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Answer 9

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Answer 10

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Answer 11

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

Solution Summary: The author proves using Kepler's Third Law and Newton’s Law of Motion and gravity that the mass of the star can be expressed as M = ( 4 2 G )

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