The force exerted by the earth on the moon is:
(Consider G=\(6 \times 10^{-11}Nm^2kg^{-2}\) ; mass of the earth M=\(6 \times 10^{24}\)kg; mass of the moon= \(8 \times 10^{22}\) kg, and distance between earth and moon is \(4 \times 10^{8}\) m).
$1.8 \times 10^{20}$ N
Understanding Gravitational Force Calculation
The question asks us to calculate the force exerted by the Earth on the Moon using the provided values for the gravitational constant (\(G\)), the mass of the Earth (\(M\)), the mass of the Moon (\(m\)), and the distance between them (\(r\)). This force is governed by Newton's Law of Universal Gravitation.
Newton's Law of Universal Gravitation
Newton's Law of Universal Gravitation states that every particle attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula is:
\( F = G \frac{Mm}{r^2} \)
Where:
- \(F\) is the gravitational force between the two bodies.
- \(G\) is the universal gravitational constant.
- \(M\) is the mass of the larger body (Earth).
- \(m\) is the mass of the smaller body (Moon).
- \(r\) is the distance between the centers of the two bodies.
Given Values
Let's list the values provided in the question:
| Gravitational Constant (\(G\)) | \(6 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2}\) |
| Mass of Earth (\(M\)) | \(6 \times 10^{24} \, \text{kg}\) |
| Mass of Moon (\(m\)) | \(8 \times 10^{22} \, \text{kg}\) |
| Distance (\(r\)) | \(4 \times 10^{8} \, \text{m}\) |
Step-by-Step Force Calculation
Now, we will substitute these values into the gravitational force formula:
- Calculate the product of the masses (\(Mm\)): \( Mm = (6 \times 10^{24} \, \text{kg}) \times (8 \times 10^{22} \, \text{kg}) \) \( Mm = (6 \times 8) \times 10^{(24+22)} \, \text{kg}^2 \) \( Mm = 48 \times 10^{46} \, \text{kg}^2 \)
- Calculate the square of the distance (\(r^2\)): \( r^2 = (4 \times 10^{8} \, \text{m})^2 \) \( r^2 = 4^2 \times (10^{8})^2 \, \text{m}^2 \) \( r^2 = 16 \times 10^{16} \, \text{m}^2 \)
- Substitute \(Mm\) and \(r^2\) into the formula part \(\frac{Mm}{r^2}\): \( \frac{Mm}{r^2} = \frac{48 \times 10^{46} \, \text{kg}^2}{16 \times 10^{16} \, \text{m}^2} \) \( \frac{Mm}{r^2} = \left(\frac{48}{16}\right) \times 10^{(46-16)} \, \frac{\text{kg}^2}{\text{m}^2} \) \( \frac{Mm}{r^2} = 3 \times 10^{30} \, \frac{\text{kg}^2}{\text{m}^2} \)
- Multiply by the gravitational constant (\(G\)) to find the force (\(F\)): \( F = G \times \frac{Mm}{r^2} \) \( F = (6 \times 10^{-11} \, \text{Nm}^2\text{kg}^{-2}) \times (3 \times 10^{30} \, \frac{\text{kg}^2}{\text{m}^2}) \) \( F = (6 \times 3) \times 10^{(-11+30)} \, \text{N} \) \( F = 18 \times 10^{19} \, \text{N} \)
- Express the final answer in standard scientific notation: \( F = 1.8 \times 10^1 \times 10^{19} \, \text{N} \) \( F = 1.8 \times 10^{20} \, \text{N} \)
Conclusion
The calculated force exerted by the Earth on the Moon is \(1.8 \times 10^{20}\) N. This matches one of the options provided.
Similar Questions
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Which of the following statements is/are correct?
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I) The mass of an object is the same as its weight.
II) The mass of an object is not the same as its weight.
III) Mass is scalar quantity, whereas weight is vector quantity.
Which of the following statements is/are correct?
Statements:
I) The Gravitational force between two objects is independent of the distance of separation between two objects.
II) The Gravitational force between two objects is inversely proportional to the square of the distance of the separation between two objects.
III) The Gravitational force between two objects is directly proportional to the product of their masses.
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Energy possessed by a body by virtue of its position or configuration is:
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A. The value of G on the moon is equal to that on the earth.
B. $26.68 \times 10^{-11}$ N is the force of gravitation between two point masses of 2 kg and 2 kg kept 1 m apart.
C. Newton's law of gravitation is valid in the laboratory only.
D. Force is inversely proportional to the square of the distance between two bodies. - What will be the force of attraction between two bodies weighing 20 kg and 50 kg, respectively, with a distance of 2 metre between them?
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