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Two bowling balls each have a mass of 8 kg. If they are 2 m apart, what is the gravitational force between them? [tex]\( G = 6.67 \times 10^{-11} \, N \cdot (m/kg)^2 \)[/tex]

A. [tex]\( 2.14 \times 10^{-9} \, N \)[/tex]
B. [tex]\( 3.21 \times 10^{-8} \, N \)[/tex]
C. [tex]\( 2.68 \times 10^{-10} \, N \)[/tex]
D. [tex]\( 1.07 \times 10^{-9} \, N \)[/tex]

Sagot :

GinnyAnswer
To find the gravitational force between two objects, you can use Newton's law of universal gravitation. The formula is:

[tex]\[ F = G \frac{m_1 m_2}{r^2} \][/tex]

where:
- [tex]\( F \)[/tex] is the gravitational force between the two objects,
- [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] and [tex]\( m_2 \)[/tex] are the masses of the two objects (in kilograms),
- [tex]\( r \)[/tex] is the distance between the centers of the two objects (in meters).

Given:
- The mass of each bowling ball, [tex]\( m_1 = m_2 = 8 \, \text{kg} \)[/tex],
- The distance between the two bowling balls, [tex]\( r = 2 \, \text{m} \)[/tex].

Now plug these values into the formula:

[tex]\[ F = \left( 6.67 \times 10^{-11} \, \text{N} \cdot \left(\text{m}^2 / \text{kg}^2\right) \right) \frac{8 \, \text{kg} \times 8 \, \text{kg}}{(2 \, \text{m})^2} \][/tex]

First, calculate [tex]\( (2 \, \text{m})^2 \)[/tex]:

[tex]\[ (2 \, \text{m})^2 = 4 \, \text{m}^2 \][/tex]

Next, calculate [tex]\( 8 \, \text{kg} \times 8 \, \text{kg} \)[/tex]:

[tex]\[ 8 \, \text{kg} \times 8 \, \text{kg} = 64 \, \text{kg}^2 \][/tex]

Now the formula looks like this:

[tex]\[ F = \left( 6.67 \times 10^{-11} \, \text{N} \cdot (\text{m}^2 / \text{kg}^2) \right) \frac{64 \, \text{kg}^2}{4 \, \text{m}^2} \][/tex]

Divide [tex]\( 64 \, \text{kg}^2 \)[/tex] by [tex]\( 4 \, \text{m}^2 \)[/tex]:

[tex]\[ \frac{64 \, \text{kg}^2}{4 \, \text{m}^2} = 16 \, \text{kg}^2 / \text{m}^2 \][/tex]

Now multiply:

[tex]\[ F = 6.67 \times 10^{-11} \, \text{N} \cdot (\text{m}^2 / \text{kg}^2) \times 16 \, \text{kg}^2 / \text{m}^2 \][/tex]

[tex]\[ F = 6.67 \times 10^{-11} \times 16 \][/tex]

Calculate [tex]\( 6.67 \times 16 \)[/tex]:

[tex]\[ 6.67 \times 16 = 106.72 \][/tex]

Now add the exponent:

[tex]\[ 106.72 \times 10^{-11} = 1.0672 \times 10^{-9} \, \text{N} \][/tex]

So, the gravitational force between the two bowling balls is:

[tex]\[ F = 1.0672 \times 10^{-9} \, \text{N} \][/tex]

Referring to the given options:

- A. [tex]\( 2.14 \times 10^{-9} \, \text{N} \)[/tex]
- B. [tex]\( 3.21 \times 10^{-8} \, \text{N} \)[/tex]
- C. [tex]\( 2.68 \times 10^{-10} \, \text{N} \)[/tex]
- D. [tex]\( 1.07 \times 10^{-9} \, \text{N} \)[/tex]

The closest and correct answer is:

D. [tex]\( 1.07 \times 10^{-9} \, \text{N} \)[/tex]
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