What happens to the gravitational force between two objects if the mass of one object is doubled and the distance between them is also doubled?
The force would be halved
Understanding Gravitational Force Changes
Let's explore how the gravitational force between two objects changes when their masses and the distance between them are altered. This involves using Newton's Law of Universal Gravitation, which describes the attractive force between any two objects with mass.
Newton's Law of Universal Gravitation
The formula for the gravitational force (\(F\)) between two objects is given by:
\(F = G \frac{m_1 m_2}{r^2}\)
Where:
- \(G\) is the gravitational constant (a fixed value).
- \(m_1\) is the mass of the first object.
- \(m_2\) is the mass of the second object.
- \(r\) is the distance between the centers of the two objects.
This formula tells us that gravitational force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.
Analyzing the Scenario: Doubling Mass and Distance
We are given an initial situation with two objects having masses \(m_1\) and \(m_2\) at a distance \(r\). The initial gravitational force is:
\(F_{\text{initial}} = G \frac{m_1 m_2}{r^2}\)
Now, consider the changes described:
- The mass of one object (let's say \(m_1\)) is doubled. So, the new mass is \(m_1' = 2m_1\).
- The distance between the objects is also doubled. So, the new distance is \(r' = 2r\).
The mass of the second object (\(m_2\)) remains unchanged.
Calculating the New Gravitational Force
Now, we can calculate the new gravitational force (\(F_{\text{new}}\)) using the modified values in the formula:
\(F_{\text{new}} = G \frac{m_1' m_2}{(r')^2}\)
Substitute the new values (\(m_1' = 2m_1\) and \(r' = 2r\)):
\(F_{\text{new}} = G \frac{(2m_1) m_2}{(2r)^2}\)
Simplify the expression:
\(F_{\text{new}} = G \frac{2m_1 m_2}{4r^2}\)
We can rearrange the terms:
\(F_{\text{new}} = \frac{2}{4} \times G \frac{m_1 m_2}{r^2}\)
\(F_{\text{new}} = \frac{1}{2} \times \left( G \frac{m_1 m_2}{r^2} \right)\)
Notice that the term inside the parentheses is the original gravitational force, \(F_{\text{initial}}\).
\(F_{\text{new}} = \frac{1}{2} F_{\text{initial}}\)
This result shows that the new gravitational force is half of the initial gravitational force.
Summary of Changes
| Factor | Initial State | Change | New State | Effect on Force Formula (\(F \propto \frac{m_1 m_2}{r^2}\)) |
|---|---|---|---|---|
| Mass 1 (\(m_1\)) | \(m_1\) | Doubled | \(2m_1\) | Multiplies force by 2 (direct proportion) |
| Mass 2 (\(m_2\)) | \(m_2\) | No change | \(m_2\) | No change in this term |
| Distance (\(r\)) | \(r\) | Doubled | \(2r\) | Divides force by \((2r)^2 = 4r^2\) (inverse square law) |
| Overall Force (\(F\)) | \(F_{\text{initial}}\) | Combined effect | \(F_{\text{new}}\) | Force multiplied by \(\frac{2 \times 1}{(2)^2} = \frac{2}{4} = \frac{1}{2}\) |
Therefore, the gravitational force between the two objects would be halved.
Conclusion on Gravitational Force
When one mass is doubled, the gravitational force tends to double. However, when the distance is doubled, the gravitational force is divided by the square of the distance change, which is \(2^2 = 4\). The combined effect is that the force changes by a factor of \(\frac{2}{4} = \frac{1}{2}\). This means the force is halved.
Revision Table: Gravitational Force Factors
| Factor Changed | How it Affects Gravitational Force (\(F \propto \frac{m_1 m_2}{r^2}\)) |
|---|---|
| Mass (\(m_1\) or \(m_2\)) is doubled | Force is doubled (\(F \propto 2\)) |
| Mass (\(m_1\) or \(m_2\)) is halved | Force is halved (\(F \propto \frac{1}{2}\)) |
| Distance (\(r\)) is doubled | Force is divided by \(2^2 = 4\) (\(F \propto \frac{1}{2^2} = \frac{1}{4}\)) |
| Distance (\(r\)) is halved | Force is multiplied by \(2^2 = 4\) (\(F \propto \frac{1}{(1/2)^2} = 4\)) |
Additional Information: Inverse Square Law in Physics
The relationship where a physical quantity is inversely proportional to the square of the distance from the source is called an inverse square law. Gravitational force follows an inverse square law with distance. Other phenomena that follow an inverse square law include:
- Electric force between point charges (Coulomb's Law).
- Intensity of light from a point source.
- Intensity of sound from a point source.
Understanding the inverse square law is crucial in physics for analyzing forces and intensities spread out over space.
Important Questions from Universal law of gravitation
Which of the following laws says that "Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them?"
The force of attraction between two objects of masses 'M' and 'm' which lie at a distance 'd' from each other is directly proportional to the-
The force of attraction (F) between two particles having masses m 1and m 2is given by _______. (If r is the distance between them and G is a universal constant)
Three point masses each of mass m are placed at the three corners of an equilateral triangle of side x. Find the resultant force acting on any one particle at the corner.
Imagine a light planet is revolving around a star in a circular orbit of radius R with the period of revolution T . If the gravitational force of attraction between the two is proportional to R(-5/2) then
