Problem 36 - Master Resource Book in JEE Main Physics

Question

Supposing Newton's law of gravitation for gravitation forces \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) between two masses \(m_{1}\) and \(m_{2}\) at positions \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) read $$ \mathbf{F}_{1}=-\mathbf{F}_{2}=-\frac{\mathbf{r}_{12}}{r_{12}^{3}} G M_{0}^{2}\left(\frac{m_{1} m_{2}}{M_{0}^{2}}\right)^{n} $$ where \(M_{0}\) is a constant of dimension of mass, \(\mathbf{r}_{12}=\mathbf{r}_{1}-\mathbf{r}_{2}\) and \(n\) is a number. In such \(a\) case (a) the acceleration due to gravity on earth will be different for different objects (b) none of the three laws of Kepler will be valid (c) only the third law will become invalid (d) for \(n\) negative, an object lighter than water will sink in water

Step-by-Step Solution

Answer

(b) none of the three laws of Kepler will be valid.

Step 1: Understanding the Given Problem

We are given a modified version of Newton's law of gravitation, which includes a factor involving an exponent \( n \). The force between two masses \( m_1 \) and \( m_2 \) is expressed as \( \mathbf{F}_{1} = -\mathbf{F}_{2} = -\frac{\mathbf{r}_{12}}{r_{12}^{3}} G M_{0}^{2}\left(\frac{m_{1} m_{2}}{M_{0}^{2}}\right)^{n} \). We need to assess the consequences of this modified law on gravity and the validity of physical laws.

Step 2: Analyzing the Effect on Accelerations due to Gravity

In standard gravity, \( F = G \frac{m_1 m_2}{r_{12}^2} \) leads to a gravitational acceleration \( a = \frac{F}{m_2} = G \frac{m_1}{r_{12}^2} \). However, if the force law modifies as given, the acceleration formula also changes but not proportionally to mass in a simple way, hence in general case (a) the acceleration changes differently for different objects, this makes option (a) true if \( n eq 0 \).

Step 3: Assessing the Validity of Kepler's Laws

Kepler's laws depend on the fact that the force is inversely proportional to the square of the distance \( F \sim \frac{1}{r^2} \). If \( n eq 0 \), the force is not strictly proportional to \( \frac{1}{r^2} \) anymore, affecting all of Kepler's laws. Hence, option (b) is true, not just (c).

Step 4: Considering the Condition for Sinking

When \( n \) is negative, the force magnitude decreases with increasing mass product \( m_1 m_2 \); this means lighter objects may experience stronger forces compared to heavier objects. Therefore, for an object lighter than water, gravity's pull could be sufficient to overcome buoyant forces, causing it to sink, making option (d) true for negative \(n\).

Step 5: Selecting the Correct Answer Based on Analysis

Analyzing options, (a), (b), and (d) appear valid under conditions provided if \( n eq 0 \) or negative. However, as changing \( n \) to negative invalidates the scaling of forces uniformly, and all options stating changes to motion or force behavior apply under respective \( n \) conditions.

Key Concepts

Gravitational Force

Gravitational force is a fundamental force of nature that draws two objects with mass towards each other. The classic Newton's law of gravitation tells us that the gravitational force \( F \) between two masses \( m_1 \) and \( m_2 \) is given by the equation:\[ F = G \frac{m_1 m_2}{r^2} \\]where \( G \) is the gravitational constant and \( r \) is the distance between the centers of the two masses. This law indicates that:

• The force is directly proportional to the product of the two masses \( m_1 \) and \( m_2 \).
• The force is inversely proportional to the square of the distance \( r \) between the two masses.

This principle means as the distance \( r \) increases, the gravitational force decreases, explaining phenomena such as why Earth stays in orbit around the Sun without crashing into it. In the original exercise, a modified version of this law was introduced, which adjusts this classic formula by incorporating an exponent \( n \), thereby changing the nature of the gravitational force significantly and affecting various physical principles.

Kepler's Laws

Kepler's laws describe the motion of planets around the sun, valuable for understanding celestial mechanics. They are fundamentally reliant on aspects of gravitational force:

• First Law: Law of Ellipses - This states that planets move in elliptical orbits with the Sun at one focus. It reflects gravitational interactions that deviate slightly from perfect circles due to the varying distances between celestial bodies.
• Second Law: Law of Equal Areas - This law specifies that a line joining a planet and the Sun sweeps out equal areas during equal intervals of time. This reflects how the speed of a planet varies depending on its distance from the Sun, due to gravitational forces.
• Third Law: Law of Harmonies - Expresses a relationship between the orbit period and distance from the Sun. It states that the square of the period of any planet is proportional to the cube of the semi-major axis of its orbit.

Kepler's laws rest on the gravitational force being inversely proportional to the square of the distance. If a relationship alters, as in the original problem's modified formula, it could invalidate Kepler's laws.

Acceleration Due to Gravity

Acceleration due to gravity, typically represented as \( g \), is the acceleration experienced by an object solely due to the gravitational pull of a massive body like Earth. On Earth, this acceleration is approximately \( 9.81 \, \text{m/s}^2 \). When calculating gravitational acceleration, we traditionally use:\[ g = G \frac{M}{r^2} \\]where \( M \) is the mass of the Earth, \( r \) the distance from the center of the Earth, and \( G \) the gravitational constant. This equation highlights how:

• Gravitational acceleration is independent of the mass of the falling object, meaning all objects fall with the same acceleration near Earth's surface if there is no air resistance.
• Gravitational acceleration decreases with increased altitude, as \( r \) gets larger farther from Earth's core.

In the exercise, modifications to the gravitational force law with the introduction of \( n \) imply changes to the typical calculations of gravitational acceleration. This results in innovations in gravitational behavior where different objects may experience varying accelerations.