Question

Statement-1: The gravitational force between a thin rod of finite length and a disc of infinite radius is inversely proportional to the square of distance between their centre of mass. Mass is uniformly distributed on both. Statement-2: Gravitational field intensity due to a unifrom disc of infinite radius is independent of distance from disc. (1) Statement- 1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement- 1 (2) Statement- 1 is True, Statement-2 is True; Statement- 2 is NOT a correct explanation for Statement-1 (3) Statement- 1 is True, Statement-2 is False (4) Statement- 1 is False, Statement-2 is True.

Short answer

Statement-1 is False, Statement-2 is True.

Step-by-step solution

Understanding Statement-1

Statement-1 says the gravitational force between a thin rod of finite length and a disc of infinite radius is inversely proportional to the square of the distance between their centers of mass. Recall that gravitational force between two masses is given by Newton's Law of Gravitation: ' ' which states force is inversely proportional to the square of distance, but in complex systems like a rod and an infinite disk, the distribution changes. For an infinite disk, field intensity changes differently, not fitting the simple inverse square relationship accurately.

Analyzing Statement-2

Statement-2 posits that the gravitational field intensity due to a uniform disc of infinite radius is independent of distance from the disc. For an infinitely large disc, the gravitational field away from the plane of the disc approaches a constant value due to the uniform mass distribution over an infinite surface area.

Determining the Truth of Statements

Both statements can be evaluated now: - Statement-1 is False since it oversimplifies the relationship and doesn't account for the unique distribution from the infinite disc. - Statement-2 is True as the gravitational field intensity indeed becomes independent of the distance from the disc.

Concluding the Explanation Relationship

Since Statement-1 is False and Statement-2 is True, the explanation test is moot. One cannot explain the other correctly.

Key concepts

Gravitational Field Intensity

Gravitational field intensity describes how strong the gravitational field is at a particular point in space. Imagine you are standing on Earth's surface; the gravitational field intensity would tell you how much force Earth's gravity exerts on a unit mass at that location.
In simpler terms, it's like measuring how 'heavy' you would feel due to gravity at a specific point. For a uniform disc of infinite radius, the gravitational field intensity becomes a fascinating subject. Contrary to smaller objects where the field drops off with distance, such an enormous disc creates a unique situation.
In the case of an infinite disc, due to its vast size and uniform mass distribution, the gravitational field intensity becomes independent of distance. As if, no matter how far you go, the gravitational intensity remains constant. This property stemmed from the fact that more matter keeps adding up as you move away, balancing out the reduction due to distance.

Newton's Law of Gravitation

Newton's Law of Gravitation is fundamental in understanding gravitational forces. This law states that any two masses attract each other 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.
The formula for this gravitational force is given by:
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Here:

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describes the gravitational force between two masses. The inverse-square nature implies that if the distance between two objects doubles, the gravitational force decreases by a factor of four.
For common objects like balls or small rods, this law applies beautifully. However, once we move to complex shapes like a finite rod and an infinite disk, this simple law needs more insight.
In such cases, the specific distribution of mass introduces extra complexity. Newton's Law still underpins the interactions, but the direct application may not be straightforward, and other considerations, like the object's geometry, come into play.

Uniform Mass Distribution

Uniform mass distribution means that the mass is spread out evenly over the object's volume. For example, a uniform rod has the same amount of matter in each segment along its length.
In our exercise, both the rod and the disc are said to have uniformly distributed mass. This is a crucial detail for determining gravitational force, as uniform distribution allows us to use symmetry in calculations.
Consider a uniform disc of infinite radius. The enormous size means each small piece of the disc contributes equally to the gravitational field. This cumulative effect results in the field intensity remaining constant, regardless of distance.
If the mass were not uniformly distributed, points closer or further from denser regions would experience varying intensities. Uniformity therefore plays a key role in simplifying many otherwise complex gravitational interactions.
This concept of uniform mass distribution, while theoretical in some cases like an infinite disc, helps to understand and solve gravitational problems in practical applications.