Problem 818 - JEE Physics

Question

Direction (Read the following questions and choose) (A) If both Assertion and Reason are true and the Reason is correct explanation of assertion (B) If both Assertion and Reason are true, but reason is not correct explanation of the Assertion (C) If Assertion is true, but the Reason is false (D) If Assertion is false, but the Reason is true Assertion: The value of acc. due to gravity \((\mathrm{g})\) does not depend upon mass of the body Reason: This follows from \(\mathrm{g}=\left[(\mathrm{GM}) / \mathrm{R}^{2}\right]\), where \(\mathrm{M}\) is mass of planet (earth) and \(\mathrm{R}\) is radius of planet (earth) (a) \(\mathrm{A}\) (b) \(\mathrm{B}\) (c) \(\mathrm{C}\) (d) D

Step-by-Step Solution

Answer

(a) A

Step 1: Review the formula for acceleration due to gravity

The formula for acceleration due to gravity is given by: \[g = \frac{GM}{R^2}\] Here, G is the gravitational constant, M is the mass of the planet (in this case, Earth), and R is the radius of the planet (Earth's radius).

Step 2: Recognize which variables are relevant for the Assertion

The Assertion claims that acceleration due to gravity (g) doesn't depend on the mass of the body experiencing the gravitational force. In the formula for acceleration due to gravity, the mass of the body is not a variable. The mass of the planet and the radius of the planet are the only variables used.

Step 3: Assess the Reason with respect to the Assertion

Given the formula for acceleration due to gravity, we can conclude that the mass of the body experiencing the gravitational force doesn't directly affect the value of g. In this sense, the Reason correctly explains the Assertion.

Step 4: Choose the correct option

Based on our analysis, both the Assertion and Reason are true, and the Reason is a correct explanation of the Assertion. Therefore, the correct answer is: (a) A

Key Concepts

Gravitational Constant

The gravitational constant, denoted as \( G \), is a fundamental value in Newton's law of universal gravitation. It is a measure of the strength of gravity at any scale. Without \( G \), we couldn't predict or measure gravitational forces between two masses, whether they're planets or pebbles. The value of \( G \) is approximately \( 6.67430 \times 10^{-11} \ \text{Nm}^2/\text{kg}^2 \). This small value indicates that gravity is a relatively weak force compared to others like electromagnetism.

• \( G \) allows us to calculate gravitational forces in the universe.
• It's constant—not changing with circumstances or time.
• Essential for predicting the movements and interactions of celestial bodies.

When understanding gravitational phenomena, \( G \) is the key that ties planetary masses and distances together in equations. It's crucial for accurate predictions and helps bridge earthly physics with cosmic scales.

Mass of the Planet

The mass of a planet, represented by \( M \), is a critical factor in determining the gravitational force it exerts on objects. For Earth, \( M \) is approximately \( 5.972 \times 10^{24} \ \text{kg} \). This immense mass generates Earth's gravitational pull, which we experience as our weight.

• The larger the mass, the stronger its gravitational force.
• It's the reason why we stay grounded on Earth.
• In the formula \( g = \frac{GM}{R^2} \), \( M \) directly influences \( g \).

Since the mass of the planet remains constant in most contexts we'll encounter, \( M \) provides a stable gravitational field, important for the consistency of gravitational acceleration on the planet’s surface.

Radius of the Planet

The radius of a planet, denoted as \( R \), is another vital component in the computation of gravitational acceleration. For Earth, the mean radius is about 6,371 kilometers. In the formula \( g = \frac{GM}{R^2} \), the radius is a crucial factor as it determines how far you are from the planet's center.

• The farther away from the center, the lesser the gravitational pull.
• \( R \) is squared in the equation, meaning changes in radius have a large impact.
• It explains why gravitational force decreases as altitude increases.

Understanding \( R \) helps us grasp why gravity feels weaker on mountains than at sea level, for instance. Thus, the radius is not just a measure of size but also a determinant of surface gravity strength.

Physical Laws of Gravity

Gravity isn't just a force; it's a fundamental principle that governs objects in the universe. These physical laws, laid down by Isaac Newton, tell us how two masses attract each other with a force proportional to their masses and inversely proportional to the square of the distance between them. These laws explain:

• Planetary orbits and satellite trajectories.
• The structure of galaxies and galaxy clusters.
• Everyday phenomena like the fall of an apple or an object staying put.

In the context of gravitational acceleration \( g \), the laws of gravity clarify why \( g \) is the same for all objects, regardless of their mass. We see this when a feather and a hammer fall at the same rate in the absence of air resistance. The predictability and uniformity of gravitational laws are foundational to physics, helping us explore both theoretical and practical realms.