Problem 671 - JEE Physics

Question

Two sphere of mass \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\) are situated in air and the gravitational force between them is \(F\). The space around the masses is now filled with liquid of specific gravity 3 . The gravitational force will now be (A) \(\mathrm{F}\) (B) \(3 \mathrm{~F}\) (C) \(\mathrm{F} / 3\) (D) \(\mathrm{F} / 9\).

Step-by-Step Solution

Answer

The new gravitational force between the masses in the liquid medium is \(\frac{F}{3}\). So, the correct answer is (C) \(\frac{F}{3}\).

Step 1: State the formula for gravitational force between two masses.

The formula for the gravitational force between two masses is given by Newton's law of gravitation: \[F = G \frac{m_1 m_2}{r^2}\] Where \(F\) is the force, \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses, and \(r\) is the distance between the centers of the masses.

Step 2: Identify the change in medium properties.

When the medium changes from air to liquid, the specific gravity of the medium is increased to 3. We need to account for this change when calculating the new gravitational force.

Step 3: Relate specific gravity to the actual gravitational force.

Specific gravity is defined as the ratio of the density of the substance to the density of a reference substance. In this case, the reference substance is air, and we assume it has a specific gravity of 1. Therefore, the force \(F\) that we have initially is the actual gravitational force between the masses in air. When the medium changes, the new medium's density will affect how the masses interact gravitationally.

Step 4: Calculate the gravitational force in the liquid medium.

When the medium is changed to a liquid with a specific gravity of 3, the actual gravitational force between the masses becomes: \[F' = \frac{F}{3}\] This is because the denser liquid medium will reduce the apparent masses of the objects due to buoyancy, making them feel lighter, and the force between them will be reduced.

Step 5: Find the correct answer among the given options.

Based on our calculations, the new gravitational force between the masses in the liquid medium is \(\frac{F}{3}\). Therefore, the correct answer is: (C) \(\frac{F}{3}\).

Key Concepts

Newton's Law of Gravitation

Newton's Law of Gravitation describes the gravitational attraction between two masses. According to this law, every point mass attracts every other point mass in the universe 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. This can be mathematically expressed as:\[F = G \frac{m_1 m_2}{r^2}\]Where:

• \(F\) is the gravitational force between the two masses
• \(G\) is the gravitational constant, approximately \(6.674 \times 10^{-11} \, \text{N}\cdot\text{m}^2/\text{kg}^2\)
• \(m_1\) and \(m_2\) are the masses of the two objects
• \(r\) is the distance between the centers of the two masses

This universal law explains the force of gravity that we experience on Earth, which keeps the planets in orbit around the Sun and governs the behavior of astronomical bodies.

Medium Change Effects

When a medium around masses changes, such as from air to a liquid, it can affect the gravitational interaction between the masses. This is primarily due to the effect of buoyancy, which can alter the apparent weight of the masses. The specific gravity of a medium is the ratio of its density compared to a reference (usually water for liquids). If the specific gravity is greater than 1, the medium is denser than the reference.

• As the medium's density increases, buoyancy effects become more prominent.
• Buoyancy can reduce the effective force of gravity as it creates an upward force opposing gravity.
• In the exercise, filling the space with a liquid of specific gravity 3 reduces the apparent gravitational force by a factor of 3 due to increased buoyancy.

Understanding how buoyancy works in different media is crucial when calculating physical forces, such as gravitational force, as seen in the problem.

Specific Gravity

Specific gravity is a dimensionless quantity that represents the ratio of the density of a substance to a reference substance. Water is commonly used as the reference substance when dealing with liquids. The formula for specific gravity \(SG\) is:\[ SG = \frac{\text{Density of the substance}}{\text{Density of the reference substance}} \]A specific gravity of 3 implies that the substance is three times denser than the reference substance, which in this case, is air.

• Specific gravity helps determine how substances interact when submerged in a fluid.
• A denser medium will increase buoyant forces acting on submerged objects, affecting calculations involving weight and gravitational interactions.
• In the given exercise, the introduction of a fluid with specific gravity 3 alters the gravitational force, illustrating the importance of considering medium properties in physics problems.

By examining specific gravity, scientists and engineers can anticipate how objects behave in different environmental conditions, both in practical applications and theoretical calculations.