Problem 812 - JEE Physics

Question

Suppose the gravitational force varies inversely as the nth power of distance the time period of planet in circular orbit of radius \(\mathrm{R}\) around the sun will be proportional to (A) \(\mathrm{R}^{[(\mathrm{n}+1) / 2]}\) (B) \(\mathrm{R}^{[(\mathrm{n}-1) / 2]}\) (C) \(\mathrm{R}^{\mathrm{n}}\) (D) \(\mathrm{R}^{[(\mathrm{n}-1) / 2]}\)

Step-by-Step Solution

Answer

The time period (T) of a planet in a circular orbit of radius R around the sun, when the gravitational force varies inversely as the nth power of distance, is proportional to \(R^{[(n+1) / 2]}\). Therefore, the correct answer is (A) \(R^{[(n+1) / 2]}\).

Step 1: Recall Kepler's Third Law

Kepler's third law states that the square of the time period of a planet's orbit around the sun is proportional to the cube of the semi-major axis of the orbit. For a circular orbit, the semi-major axis is equal to the radius of the orbit. Mathematically, this can be written as: \(T^2 = k R^3\), where T is the time period, R is the radius of the orbit, and k is the constant of proportionality.

Step 2: Modify Kepler's Third Law for the given force variation

According to the problem, the gravitational force varies inversely as the nth power of distance, which can be written as: \(F = \frac{G M m}{R^n}\), where F is the gravitational force, G is the gravitational constant, M is the mass of the sun, m is the mass of the planet, and n is the power of distance dependence.

Step 3: Express centripetal force in terms of radius

Centripetal force is required for the planet to stay in a circular orbit around the sun. This force is equal to the gravitational force. Therefore, we can write: \(\frac{G M m}{R^n} = \frac{m v^2}{R}\), where v is the tangential velocity of the planet. Now, we can rearrange this equation to find the relationship between v and R: \(v^2 = \frac{G M R^{(n-2)}}{R^n} = G M R^{(2-n)}\).

Step 4: Relate time period, velocity and radius

The time period can also be defined as the time taken for the planet to complete one full orbit around the sun, which can be defined as: \(T = \frac{2 \pi R}{v}\), Now, square T to find the relation between T and R: \(T^2 = (\frac{2 \pi R}{v})^2\).

Step 5: Derive the relationship between time period and radius

Now, substitute the expression for v^2 from step 3 into the equation for T^2 from step 4: \(T^2 = (\frac{2 \pi R}{\sqrt{G M R^{(2-n)}}})^2\) \(T^2 = \frac{4 \pi^2 R^2}{G M R^{(2-n)}}\), Then, simplify to find the relationship between T and R: \(T^2 = \frac{4 \pi^2}{G M} R^{(n+1)}\) Hence, \(T \propto R^{[(n+1)/2]}\).

Step 6: Find the correct answer

Comparing the expression found in step 5 with the given options, we can conclude that the correct answer is: (A) \(R^{[(n+1) / 2]}\).

Key Concepts

Gravitational Force

Gravitational force is a fundamental force of nature that acts between two masses. Imagine it as a pulling force, keeping planets in their orbits and people grounded on Earth. It is always attractive, and it acts over vast distances in the universe. According to Newton's law of universal gravitation, this force is directly proportional to the masses of the objects and inversely proportional to the square of the distance between them. In mathematical terms, it's expressed as:

• \( F = \frac{G M m}{R^2} \)

Here, \( F \) is the gravitational force, \( G \) is the gravitational constant, \( M \) and \( m \) are the masses of the two objects, and \( R \) is the distance between their centers. In the problem, this gravitational force is modified to vary inversely as the *n*th power of distance, adjusting our understanding of how distance affects gravitational attraction.

Circular Orbits

Picture a planet moving around the sun in a path that is perfectly circular. A circular orbit means that the distance (or radius \( R \)) between the planet and the sun remains constant. Such an orbit provides a simplified scenario to analyze celestial mechanics. The centripetal force required to maintain this circular path is supplied by gravitational force. Therefore, the centripetal force equation becomes vital:

• \( \frac{m v^2}{R} = F \)

Here, \( v \) is the tangential velocity of the planet, which helps in connecting velocity to gravitational force, which is crucial when formulating the time period of the planet's orbit. The beauty of circular orbits lies in this balance between gravitational and centripetal forces.

Time Period of Planets

The time period of a planet in orbit, often seen as \( T \), is the time it takes for a planet to complete one full orbit around the sun. Kepler’s third law traditionally relates this time period to the semi-major axis of the orbit (for circular paths, this is the radius \( R \)) with the relation:

• \( T^2 \propto R^3 \)

In this formula, the square of the orbital period \( T^2 \) is proportional to the cube of the orbit's radius. However, when gravitational force changes with distance to the *n*th power, this relationship needs adjustment. By analyzing the dynamics of how force and velocity contribute to the orbital time, we derive that:

• \( T^2 = \frac{4 \pi^2}{G M} R^{(n+1)} \)
• Simplifying further, \( T \propto R^{[(n+1)/2]} \)

Thus, the time period becomes dependent on the modified power of radius, shaped by the force's inverse dependence on distance.

Inverse Power Law of Distance

The inverse power law of distance is an important concept when modifying gravitational laws. Normally, the gravitational force is inversely proportional to the square of the distance, represented as \( R^2 \). However, the problem introduces an alternate scenario where the force varies inversely as the \( n \)th power of the distance:

• \( F = \frac{G M m}{R^n} \)

This change affects how the force decreases as distance increases. As \( n \) changes, the rate at which gravitational pull diminishes varies. For example, if \( n \) increases, the pull weakens much faster with distance. This adjustment has cascading effects on orbital mechanics, influencing everything from centripetal force equations to orbital periods. Such modifications help physicists explore theoretical models spanning various cosmic contexts.