If the distance between two objects is increased by two times, the gravitational force between them will
decrease by four times
Understanding Gravitational Force and Distance
The question asks how the gravitational force between two objects changes when the distance between them is increased. This involves understanding Newton's Law of Universal Gravitation, which describes the force of attraction between any two objects with mass.
Newton's Law of Universal Gravitation
Sir Isaac Newton formulated the law that states every particle in the universe attracts every other particle 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 mathematical representation of this law is:
\[F = G \frac{m_1 m_2}{r^2}\]
Where:
- \(F\) is the gravitational force between the two objects.
- \(G\) is the universal gravitational constant.
- \(m_1\) and \(m_2\) are the masses of the two objects.
- \(r\) is the distance between the centers of the two objects.
The Relationship Between Force and Distance
From the formula, we can see how the gravitational force \(F\) depends on the distance \(r\). The force is inversely proportional to the square of the distance (\(F \propto \frac{1}{r^2}\)). This relationship is known as the inverse square law.
This means if the distance increases, the force decreases, and if the distance decreases, the force increases. The change is not linear; it depends on the square of the distance change.
Calculating the Change in Gravitational Force
Let's consider the initial situation and the new situation where the distance is increased.
- Let the initial distance between the two objects be \(r\).
- The initial gravitational force is \(F_{\text{initial}} = G \frac{m_1 m_2}{r^2}\).
Now, the distance between the two objects is increased by two times. This means the new distance, let's call it \(r_{\text{new}}\), is \(2r\).
- The new distance is \(r_{\text{new}} = 2r\).
Now we can calculate the new gravitational force, \(F_{\text{new}}\), using the new distance \(r_{\text{new}}\):
\[F_{\text{new}} = G \frac{m_1 m_2}{(r_{\text{new}})^2}\]
Substitute the new distance \(r_{\text{new}} = 2r\) into the equation:
\[F_{\text{new}} = G \frac{m_1 m_2}{(2r)^2}\]
Simplify the denominator:
\[F_{\text{new}} = G \frac{m_1 m_2}{4r^2}\]
We can rewrite this expression by separating the factor of \(1/4\):
\[F_{\text{new}} = \frac{1}{4} \left(G \frac{m_1 m_2}{r^2}\right)\]
Notice that the term in the parentheses, \(G \frac{m_1 m_2}{r^2}\), is the initial gravitational force, \(F_{\text{initial}}\).
So, we have:
\[F_{\text{new}} = \frac{1}{4} F_{\text{initial}}\]
Conclusion on Force Change
The new gravitational force is one-fourth of the initial gravitational force. This means the force has decreased. Specifically, it has decreased by a factor of 4.
Therefore, if the distance between two objects is increased by two times, the gravitational force between them will decrease by four times.
Revision Table: Gravitational Force vs. Distance
| Change in Distance (Factor) | Change in Force (Factor) | Effect on Force |
| \(r\) | \(F\) | Original Force |
| \(2r\) | \(F \times (1/2)^2 = F/4\) | Decreases by 4 times |
| \(3r\) | \(F \times (1/3)^2 = F/9\) | Decreases by 9 times |
| \(r/2\) | \(F \times (1/(1/2))^2 = F \times 2^2 = 4F\) | Increases by 4 times |
| \(r/3\) | \(F \times (1/(1/3))^2 = F \times 3^2 = 9F\) | Increases by 9 times |
Additional Information: Gravitational Force Properties
Here are some more important points about gravitational force:
- Always Attractive: Gravitational force is always a force of attraction between objects.
- Depends on Mass: The greater the masses of the objects, the stronger the gravitational force between them. It is directly proportional to the product of the masses.
- Inverse Square Law: As demonstrated, the force decreases rapidly as the distance between objects increases due to the inverse square relationship.
- Acts at a Distance: Gravitational force does not require the objects to be in contact. It acts over any distance, although it becomes very weak over large distances due to the inverse square law.
- Weakest Fundamental Force: Gravity is the weakest of the four fundamental forces in nature (gravitational, electromagnetic, weak nuclear, strong nuclear). However, it is the dominant force on a large scale because it is always attractive and acts over infinite distances, unlike the nuclear forces which are short-range, or electromagnetic forces which can be attractive or repulsive.
Important Questions from Gravitation
Which of the following forces is responsible for the tides, due to the Moon and the Sun?
The weight of an object was 60 N when measured on the surface of the earth. What would be its weight when measured on the surface of the moon?
Seven people, A, B, C, L, X, Y, and Z are sitting in a row, facing north. No one sits to the right of Y. Only three people sit between Y and C. Only two people sit between C and Z. B sits third to the left of X. L sits to the immediate right of X.
How many people sit between A and Z?
The universal constant of gravitation G has the unit
- An object's apparent weight is slightly less at the Earth's equator compared to its poles. This difference is primarily attributed to:
