Calculating Gravitational Force Between Bodies

The question asks us to compare the gravitational force between two pairs of bodies based on their masses and the distance separating them. We will use Newton's Law of Universal Gravitation to solve this problem.

Understanding Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that the gravitational force (\(F\)) between two point masses (\(m_1\) and \(m_2\)) is directly proportional to the product of their masses and inversely proportional to the square of the distance (\(r\)) between their centers. Mathematically, the formula is:

$$F = G \frac{m_1 m_2}{r^2}$$

Here, \(G\) is the gravitational constant.

Analyzing the First System

In the first system, we have two bodies, each with mass \(M\). The distance between them is \(R\). The gravitational force between them is given as \(F\).

Using the formula, the gravitational force in the first system (\(F_1\)) is:

$$F_1 = G \frac{M \cdot M}{R^2}$$

$$F_1 = G \frac{M^2}{R^2}$$

We are told that \(F_1 = F\). So, we can write:

$$F = G \frac{M^2}{R^2} \quad \text{(Equation 1)}$$

Analyzing the Second System

In the second system, we have two bodies, each with mass \(2M\). The distance between them is \(R/2\). Let the gravitational force in this system be \(F_2\).

Using the gravitational force formula for the second system:

The masses are \(m_1' = 2M\) and \(m_2' = 2M\).

The distance is \(r' = R/2\).

$$F_2 = G \frac{m_1' m_2'}{r'^2}$$

Substitute the values of the masses and distance:

$$F_2 = G \frac{(2M)(2M)}{(R/2)^2}$$

Simplify the expression:

$$F_2 = G \frac{4M^2}{R^2/4}$$

To simplify further, invert the denominator and multiply:

$$F_2 = G \frac{4M^2}{1} \cdot \frac{4}{R^2}$$

$$F_2 = G \frac{16M^2}{R^2}$$

Comparing Forces in Both Systems

Now, we want to express \(F_2\) in terms of \(F\). We have the expression for \(F_2\):

$$F_2 = G \frac{16M^2}{R^2}$$

We can factor out 16 from the expression:

$$F_2 = 16 \left( G \frac{M^2}{R^2} \right)$$

From Equation 1, we know that \(F = G \frac{M^2}{R^2}\). Substitute \(F\) into the expression for \(F_2\):

$$F_2 = 16 F$$

Thus, the gravitational force between the bodies in the second system is 16 times the gravitational force in the first system.

Summary of Calculations

Comparing \(F_2\) and \(F\):

$$F_2 = G \frac{16M^2}{R^2} = 16 \left( G \frac{M^2}{R^2} \right) = 16 F$$

Conclusion

The gravitational force between the bodies in the second system is \(16F\).

Revision Table: Gravitational Force Concepts

Additional Information: Factors Affecting Gravitational Force

The gravitational force is a fundamental force of nature that attracts any two objects with mass towards each other. The strength of this force depends primarily on two factors:

• Mass of the Objects: The greater the mass of the objects, the stronger the gravitational pull between them. The force is directly proportional to the product of their masses. Doubling one mass doubles the force, doubling both masses quadruples the force.
• Distance Between the Objects: The distance between the centers of the objects significantly affects the gravitational force. This relationship follows an inverse square law. If the distance is doubled, the force becomes four times weaker (\(1/2^2 = 1/4\)). If the distance is halved, the force becomes four times stronger (\(1/(1/2)^2 = 1/(1/4) = 4\)).

In this problem, both the masses and the distance were changed simultaneously, leading to a combined effect on the gravitational force.