Suppose the force of gravitation between two bodies of equal masses is F. If each mass is doubled keeping the distance of separation between them unchanged, the force would become
4F
Understanding Gravitational Force
The force of gravitation between any two bodies having mass is described by Newton's Law of Gravitation. This law states that the force is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, the gravitational force $F$ between two bodies with masses $m_1$ and $m_2$ separated by a distance $r$ is given by:
$F = G \frac{m_1 m_2}{r^2}$
Here, $G$ is the universal gravitational constant.
Analyzing the Initial Gravitational Force
We are given that initially, we have two bodies of equal masses. Let's denote the mass of each body as $m$. The distance of separation between them is given as $r$. The force of gravitation between them is given as $F$.
Using the formula for gravitational force:
$F = G \frac{m \times m}{r^2}$
$F = G \frac{m^2}{r^2}$
This equation represents the initial gravitational force.
Calculating the New Gravitational Force
Now, the problem states that each mass is doubled, while the distance of separation remains unchanged.
- New mass of the first body, $m_1' = 2m$
- New mass of the second body, $m_2' = 2m$
- New distance of separation, $r' = r$ (unchanged)
Let the new gravitational force be $F'$. Using Newton's Law of Gravitation with the new values:
$F' = G \frac{m_1' m_2'}{(r')^2}$
Substitute the new values:
$F' = G \frac{(2m) \times (2m)}{(r)^2}$
$F' = G \frac{4m^2}{r^2}$
Comparing Initial and New Gravitational Force
We can rewrite the expression for the new force $F'$ by factoring out the constant 4:
$F' = 4 \times \left( G \frac{m^2}{r^2} \right)$
From our calculation of the initial force, we know that $F = G \frac{m^2}{r^2}$.
Substituting $F$ into the equation for $F'$:
$F' = 4F$
This shows that when each mass is doubled and the distance remains unchanged, the gravitational force becomes four times the original force.
Summary of Changes in Gravitational Force
Here is a summary of the parameters and forces:
| Parameter | Initial State | New State | Change |
|---|---|---|---|
| Mass 1 | $m$ | $2m$ | Doubled |
| Mass 2 | $m$ | $2m$ | Doubled |
| Distance | $r$ | $r$ | Unchanged |
| Gravitational Force | $F = G \frac{m^2}{r^2}$ | $F' = G \frac{(2m)(2m)}{r^2} = G \frac{4m^2}{r^2}$ | Becomes $4F$ |
Therefore, if the force of gravitation between two bodies of equal masses is $F$, and each mass is doubled keeping the distance of separation between them unchanged, the force would become $4F$.
Revision Table: Gravitational Force Calculation
| Concept | Formula/Principle | Application in Problem |
|---|---|---|
| Newton's Law of Gravitation | $F = G \frac{m_1 m_2}{r^2}$ | Used to express initial and new gravitational force. |
| Effect of Changing Mass | Force $\propto m_1 m_2$ | Doubling both masses ($m_1 \to 2m_1$, $m_2 \to 2m_2$) leads to force $\propto (2m_1)(2m_2) = 4m_1m_2$, a factor of 4 increase. |
| Effect of Changing Distance | Force $\propto \frac{1}{r^2}$ | Distance unchanged ($r \to r$), so no change in force due to distance. |
| Combined Effect | Multiply change factors | Change factor from masses = 4; Change factor from distance = 1. Total change factor = $4 \times 1 = 4$. New force $= 4F$. |
Additional Information: Factors Affecting Gravitational Force
The gravitational force between two objects depends on two primary factors:
- Mass: The gravitational force is directly proportional to the product of the masses of the two objects. This means if you increase the mass of either object (or both), the gravitational force between them increases. If you double one mass, the force doubles. If you double both masses, as in this problem, the force quadruples ($2 \times 2 = 4$).
- Distance: The gravitational force is inversely proportional to the square of the distance between the centers of the two objects. This means if you increase the distance, the force decreases rapidly. For example, if you double the distance, the force becomes one-fourth ($1/2^2 = 1/4$) of the original force. Conversely, if you halve the distance, the force becomes four times ($1/(1/2)^2 = 1/(1/4) = 4$) the original force.
The universal gravitational constant ($G$) is a fundamental constant that determines the strength of the gravitational interaction. Its value is approximately $6.674 \times 10^{-11} \, \text{N} \cdot (\text{m/kg})^2$.
Important Questions from Gravitation
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