Problem 61 - Master Resource Book in JEE Main Physics

Question

If the distance between the sun and the earth is increased by three times, then attraction between two will (a) remain constant (b) decrease by \(63 \%\) (c) increase by \(63 \%\) (d) decrease by \(89 \%\)

Step-by-Step Solution

Answer

The attraction decreases by 89%, so the answer is (d).

Step 1: Understanding the Gravitational Force

The gravitational attraction between two masses, like the Earth and the Sun, is given by Newton's Law of Universal Gravitation: \[ F = \frac{G M m}{r^2} \]where \(F\) is the gravitational force, \(G\) is the gravitational constant, \(M\) and \(m\) are the masses of the two bodies, and \(r\) is the distance between the centers of the two bodies. This implies that the force is inversely proportional to the square of the distance between them.

Step 2: Calculating New Distance

With the distance between Earth and Sun increased by three times, the new distance \(r'\) becomes:\[ r' = 3r \]

Step 3: Comparing the New Force to the Original Force

The new gravitational force \(F'\) with the modified distance is given by:\[ F' = \frac{G M m}{(3r)^2} = \frac{G M m}{9r^2} \]This shows that the new force \(F'\) is one-ninth of the original force \(F\), i.e.,\[ F' = \frac{1}{9}F \]

Step 4: Calculating Change in Force

The force decreases to one-ninth of its original value, which means a decrease. To express this decrease in percentage:\[ \text{Decrease percentage} = \left(1 - \frac{1}{9}\right) \times 100\% = \frac{8}{9} \times 100\% \approx 88.89\% \]

Step 5: Final Answer Formulation

Based on the calculations, the force decreases by approximately \(89\%\). Therefore, the correct answer is option (d) 'decrease by \(89\%\)'.

Key Concepts

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation is a fundamental principle that describes the gravitational attraction between two bodies. This law states that every point mass attracts every other point mass by a force pointing along the line intersecting both points. The formula for this gravitational force is: \[ F = \frac{G M m}{r^2} \]Here:

• \(F\) is the gravitational force between the two bodies.
• \(G\) is the gravitational constant, a universal value.
• \(M\) and \(m\) are the masses of the two celestial bodies, like the Earth and the Sun.
• \(r\) is the distance between the centers of these two bodies.

This means the force decreases as the distance \(r\) increases. Newton's insight into this law helps us understand not only planetary motion but also predicts how objects interact throughout the universe.

Mass and Distance Relationship

The gravitational force depends heavily on both the mass of the objects involved and the distance between them. Mass and distance have a direct impact on the magnitude of gravitational force:

• Greater masses exert a stronger gravitational pull.
• Increased distance reduces the gravitational force.

As seen in Newton's formula, increasing the mass \((M\) or \(m)\) increases the force of attraction. Conversely, increasing the distance \((r)\) diminishes it. The interaction of mass and distance highlights why celestial bodies exhibit diverse gravitational forces. Larger planets with more massive bodies or closer proximity exert stronger gravitational attractions. This principle is crucial when studying the dynamic interactions and movements in our solar system.

Inverse Square Law

The concept of the inverse square law is essential to understanding gravitational forces. According to this principle, the gravitational force between two objects diminishes with the square of the distance. Mathematically, this means: \[ F \propto \frac{1}{r^2} \]Thus, if the distance \(r\) doubles, the gravitational force is reduced to one-fourth. If the distance triples, as in our original exercise, it leads to: \[ F' = \frac{1}{9}F \]This reduction illustrates the significant impact distance has on gravitational force. The inverse square law appears in many physical phenomena, from electromagnetism to sound intensity, showcasing its broader implications across various scientific disciplines.

Percentage Change in Force

When the distance between two bodies changes, calculating the percentage change in gravitational force can help understand the extent of this alteration. From the original exercise, when the distance tripled:

• The gravitational force became one-ninth of the original force, \( F' = \frac{1}{9}F \).
• This means the force reduced by \(88.89\%\).

To understand the percentage conceptually, it involves finding how much the new force deviates from the original force: \[ \text{Decrease percentage} = \left(1 - \frac{1}{9}\right) \times 100\% = 88.89\% \]This calculation offers insight into how a small change in distance can lead to a significant reduction in force. This approach aids students in visualizing how values change in real-world applications, reinforcing the importance of understanding gravitational dynamics.