Chapter 5.1, Problem 41PS

Solution

(a)

To determine: An equation that gives $F$ in terms of $m_1,m_2,d\text{ and }G$.

The required equation is $F=\frac{G{m}_1{m}_2}{d^2}$.

Given information:

The law of universal gravitation states that the gravitational force $F$ (in Newtons) between two objects varies jointly with their masses $m_1\text{ and }{m}_2$ (in kilograms) and inversely with the square of the distance $d$ (in meters) between the two objects. The constant of variation is denoted by $G$ and is called the universal gravitational constant.

Formula used:

Two variables $x\text{ and }y$ show inverse variation if they are related as below:

  $y=\frac{a}{x}$.

Two variables $x\text{ and }y$ show direct variation if they are related as below:

  $y=ax$.

  $z$ varies jointly with $x\text{ and }y$ is shown as:

  $z=axy$.

Where, $a$ is a constant.

Explanation:

The gravitational force $F$ between two objects varies jointly with their masses $m_1\text{ and }{m}_2$.

  $F=G{m}_1{m}_2$ …… (1)

The gravitational force $F$ between two objects varies inversely with the square of the distance $d$ between the two objects.

  $F=\frac{G}{d^2}$ …… (2)

From (1) and (2),

  $F=\frac{G{m}_1{m}_2}{d^2}$

Therefore, the required equation is $F=\frac{G{m}_1{m}_2}{d^2}$.

(b)

To determine: The universal gravitational constant $G$ from the information given about Earth and the Sun.

  $G=6.7\times {10}^{-11}$

Given information:

The law of universal gravitation states that the gravitational force $F$ (in Newtons) between two objects varies jointly with their masses $m_1\text{ and }{m}_2$ (in kilograms) and inversely with the square of the distance $d$ (in meters) between the two objects. The constant of variation is denoted by $G$ and is called the universal gravitational constant. Further it is given that, mass of Earth $m_1=5.98\times {10}^{24}\text{ kg}$ , mass of Sun $m_2=1.99\times {10}^{30}\text{ kg}$ , $F=3.53\times {10}^{22}\text{ N}$ , and $d=1.50\times {10}^{11}\text{ m}$.

Formula used:

Two variables $x\text{ and }y$ show inverse variation if they are related as below:

  $y=\frac{a}{x}$.

Two variables $x\text{ and }y$ show direct variation if they are related as below:

  $y=ax$.

  $z$ varies jointly with $x\text{ and }y$ is shown as:

  $z=axy$.

Where, $a$ is a constant.

Explanation:

Consider the equation $F=\frac{G{m}_1{m}_2}{d^2}$.

Substitute $m_1=5.98\times {10}^{24}\text{ kg}$ , $m_2=1.99\times {10}^{30}\text{ kg}$ , $F=3.53\times {10}^{22}\text{ N}$ , and $d=1.50\times {10}^{11}\text{ m}$.

  $\begin{array}{c}3.53\times {10}^{22}=\frac{G\left(5.98\times {10}^{24}\right)\left(1.99\times {10}^{30}\right)}{{\left(1.50\times {10}^{11}\right)}^2}\\ G=\frac{\left(3.53\times {10}^{22}\right)\left(2.25\times {10}^{22}\right)}{\left(5.98\times {10}^{24}\right)\left(1.99\times {10}^{30}\right)}\\ \approx 6.7\times {10}^{-11}\end{array}$

Therefore, $G=6.7\times {10}^{-11}$.

(c)

To determine: The effect on gravitational force if the masses of the two objects increase and the distance between them is held constant and if the masses of the two objects are held constant and the distance between them increases.

The gravitational force increases if the masses of the two objects increase and the distance between them is held constant and decreases if the masses of the two objects are held constant and the distance between them increases.

Given information:

The law of universal gravitation states that the gravitational force $F$ (in Newtons) between two objects varies jointly with their masses $m_1\text{ and }{m}_2$ (in kilograms) and inversely with the square of the distance $d$ (in meters) between the two objects. The constant of variation is denoted by $G$ and is called the universal gravitational constant.

Formula used:

Two variables $x\text{ and }y$ show inverse variation if they are related as below:

  $y=\frac{a}{x}$.

Two variables $x\text{ and }y$ show direct variation if they are related as below:

  $y=ax$.

  $z$ varies jointly with $x\text{ and }y$ is shown as:

  $z=axy$.

Where, $a$ is a constant.

Explanation:

The gravitational force between two objects varies jointly with their masses so, if the masses of the two objects increase and the distance between them is held constant then the gravitational force will also increase.

The gravitational force between two objects varies inversely with the square of the distance between the two objects so, if the masses of the two objects are held constant and the distance between them increases then the gravitational force will decrease.

Therefore, the gravitational force increases if the masses of the two objects increase and the distance between them is held constant and decreases if the masses of the two objects are held constant and the distance between them increases.