Problem 34 - Master Resource Book in JEE Main Physics

Question

If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube law (a) planets will not have elliptic orbits (b) circular orbits of planets is not possible (c) projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic (d) there will be no gravitational force inside a spherical shell of uniform density

Step-by-Step Solution

Answer

All options (a), (b), (c), and (d) are correct.

Step 1: Understanding the Inverse-Square Law

The law of gravitation as we know it states that the gravitational force between two objects is inversely proportional to the square of the distance between their centers. Mathematically, it is expressed as \( F = \frac{Gm_1m_2}{r^2} \). This forms the foundation for the elliptic orbits of planets according to Kepler's laws.

Step 2: Hypothetical Scenario: Inverse-Cube Law

In the given problem, we consider a scenario where the gravitational force is instead inversely proportional to the cube of the distance, \( F' = \frac{Gm_1m_2}{r^3} \). This change fundamentally alters the dynamics of gravitational interactions.

Step 3: Exploring (a) Planets' Orbits

If gravitational force follows an inverse-cube law, the stable, closed orbits such as ellipses, which rely on the conservation of angular momentum and energy in an inverse-square field, are no longer supported. Thus, (a), that planets will not have elliptic orbits, is true.

Step 4: Evaluating (b) Circular Orbits

For circular orbits, a specific balance between gravitational force and centripetal force is required, which is a condition anchored in the inverse-square relationship. With an inverse-cube dependency, this balance is compromised, making circular orbits implausible, making (b) true.

Step 5: Assessing (c) Projectile Motion

Even under an inverse-cube law, on Earth's surface the gravitational field can be considered uniform due to proximity. Thus, projectile motion, generally parabolic under a constant gravitational field, approximates as such, making (c) true.

Step 6: Investigating (d) Inside a Spherical Shell

From Gauss's law for gravity, which holds except in superficial force laws, the gravitational field inside a symmetric shell is zero even under an inverse-cube law. Therefore, (d) remains true.

Key Concepts

Inverse-Square Law

The inverse-square law is a fundamental principle in physics that describes how a force, such as gravity or light intensity, is distributed over a distance. For gravity, it specifically means that the gravitational force ( \( F \) ) between two masses is proportional to the product of their masses and inversely proportional to the square of the distance ( \( r \) ) between their centers. This relationship is mathematically expressed as: \[ F = \frac{Gm_1m_2}{r^2} \]where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses of the objects, and \( r \) is the distance between the centers of the two masses. This law is crucial for understanding the motion of celestial bodies.

• It explains why planets move in predictable orbits and why artificial satellites stay in orbit.
• It also underlies many aspects of astrophysics and cosmology.

Elliptic Orbits

Elliptic orbits are the paths that planets and other celestial bodies follow around their star or host. These orbits are typically oval-shaped due to the gravitational attraction that implements the inverse-square law. According to Kepler's First Law , orbits of planets are ellipses, with the star at one of the foci.
These stable and predictable pathways result from the gravitational balance between the planet's velocity and the attraction exerted by the star. This conservation of angular momentum is vital due to the inverse-square nature of gravity. If this law were to change to an inverse-cube law, as posed in a hypothetical scenario, elliptic orbits would no longer exist.

• The balance that maintains the orbit would be disrupted.
• Celestial bodies might instead fly off or spiral into their central star.
• The harmonious dance of planets would lose its rhythm.

Centripetal Force

Centripetal force is essential for any object moving along a circular path. This inward force is required to maintain the object's circular motion. In celestial mechanics, gravity acts as the centripetal force that keeps planets in their orbits.
The condition for a stable circular orbit is reliant on the inverse-square law. For an object to maintain a circular orbit, the gravitational pull (acting as centripetal force) must exactly balance the object's inertia. In mathematical terms, it requires: \[ m \cdot v^2 / r = \frac{Gm \cdot M}{r^2} \]where \( m \) is the mass of the orbiting object, \( v \) is its velocity, \( M \) is the mass of the central body, and \( r \) is the radius of the orbit.
If the gravitational law switches to an inverse-cube relationship, this essential balance is thrown off, rendering circular orbits impossible.

• Without the inverse-square dependency, bodies cannot sustain their circular paths.
• Ensures the delicate equilibrium is maintained, allowing moons, planets, and satellites to orbit.

Gauss's Law for Gravity

Gauss's law for gravity is a way of relating gravitational fields to the distribution of mass causing them. Similar in form to Gauss's law for electricity, it is used to compute the gravitational field in situations of high symmetry. It makes use of the gravitational flux, which is the integral of the gravitational field over a closed surface.
For a spherical shell of uniform density, Gauss's law is particularly useful. It shows that inside a spherical shell, the gravitational field is zero, due to the symmetrical cancellation of gravitational forces. This is true whether we consider the inverse-square law or shift to an inverse-cube law.

• This principle explains the gravitational null within hollow spheres, like some planetary interiors.
• Indicates how gravitational forces interact in complex systems.
• Maintains its validity under different force laws.