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

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

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

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



Answer :

To determine the gravitational force between you and your textbook, we will apply Newton's law of gravitation, which states that the gravitational force [tex]\( F_{\text{gravity}} \)[/tex] between two masses is given by:

[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 the mass of the first object (in this case, your mass, [tex]\( 72 \, \text{kg} \)[/tex]).
- [tex]\( m_2 \)[/tex] is the mass of the second object (in this case, the mass of the textbook, [tex]\( 3.7 \, \text{kg} \)[/tex]).
- [tex]\( r \)[/tex] is the distance between the centers of the two objects (in this case, [tex]\( 0.33 \, \text{m} \)[/tex]).

Let's substitute the given values into the formula:

[tex]\[ F_{\text{gravity}} = \frac{6.67 \times 10^{-11} \cdot 72 \cdot 3.7}{0.33^2} \][/tex]

First, calculate the denominator:

[tex]\[ 0.33^2 = 0.1089 \][/tex]

Next, substitute this value back into the equation:

[tex]\[ F_{\text{gravity}} = \frac{6.67 \times 10^{-11} \cdot 72 \cdot 3.7}{0.1089} \][/tex]

Now, perform the multiplication in the numerator:

[tex]\[ 6.67 \times 10^{-11} \cdot 72 \cdot 3.7 = 1.779816 \times 10^{-8} \][/tex]

Then, divide by [tex]\( 0.1089 \)[/tex]:

[tex]\[ F_{\text{gravity}} = \frac{1.779816 \times 10^{-8}}{0.1089} \][/tex]

By performing the division, we get:

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

Thus, the correct answer is:

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

The option corresponding to this value is:

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

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