If your mass is 72 kg, your textbook's mass is 3.7 kg, and you and your textbook are separated by a distance of 0.33 m, what is the gravitational force between you and your textbook?

Newton's law of gravitation is:

[tex]\[ F_{\text{gravity}} = \frac{G m_1 m_2}{r^2} \][/tex]

The gravitational constant [tex]\( G \)[/tex] is [tex]\( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex].

A. [tex]\( 5.38 \times 10^{-8} \, \text{N} \)[/tex]

B. [tex]\( 1.63 \times 10^{-7} \, \text{N} \)[/tex]

C. [tex]\( 2.45 \times 10^3 \, \text{N} \)[/tex]

D. [tex]\( 4.94 \times 10^{-7} \, \text{N} \)[/tex]



Answer :

To determine the gravitational force between you and your textbook, we'll use Newton's Law of Gravitation, which is given by the formula:

[tex]\[ F_{\text{gravity}} = \frac{G \cdot m_1 \cdot m_2}{r^2} \][/tex]

Where:
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \)[/tex]
- [tex]\( m_1 \)[/tex] is your mass, [tex]\( 72 \, \text{kg} \)[/tex]
- [tex]\( m_2 \)[/tex] is the mass of the textbook, [tex]\( 3.7 \, \text{kg} \)[/tex]
- [tex]\( r \)[/tex] is the distance between you and the textbook, [tex]\( 0.33 \, \text{m} \)[/tex]

Now we'll plug these values into the formula step-by-step.

1. Calculate the product of the masses:

[tex]\[ m_1 \times m_2 = 72 \, \text{kg} \times 3.7 \, \text{kg} = 266.4 \, \text{kg}^2 \][/tex]

2. Calculate the square of the distance:

[tex]\[ r^2 = (0.33 \, \text{m})^2 = 0.1089 \, \text{m}^2 \][/tex]

3. Plug into the formula:

[tex]\[ F_{\text{gravity}} = \frac{6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times 266.4 \, \text{kg}^2}{0.1089 \, \text{m}^2} \][/tex]

4. Multiply the constant G by the product of the masses:

[tex]\[ 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2/\text{kg}^2 \times 266.4 \, \text{kg}^2 = 1.776408 \times 10^{-8} \, \text{N} \cdot \text{m}^2 \][/tex]

5. Divide by the square of the distance:

[tex]\[ \frac{1.776408 \times 10^{-8} \, \text{N} \cdot \text{m}^2}{0.1089 \, \text{m}^2} \approx 1.631669 \times 10^{-7} \, \text{N} \][/tex]

Therefore, the gravitational force between you and your textbook is approximately:

[tex]\[ 1.63 \times 10^{-7} \, \text{N} \][/tex]

So the correct answer is:

[tex]\[ \text{B. } 1.63 \times 10^{-7} \, \text{N} \][/tex]