Problem 57 - Master Resource Book in JEE Main Physics

Question

The height at which the acceleration due to gravity becomes \(g / 9\) (where \(g\) is acceleration due to gravity on the surface of the earth) in terms of \(R\), (the radius of the earth) is (a) \(R / \sqrt{2}\) (b) \(R / 2\) (c) \(\sqrt{2} R\) (d) \(2 R\)

Step-by-Step Solution

Answer

The correct height is \( 2R \), option (d).

Step 1: Identify the formula

We know that the gravitational force decreases with the square of the distance from the center of the Earth. The formula for acceleration due to gravity at a height \( h \) above the surface of the Earth is \( g' = \frac{g R^2}{(R + h)^2} \) where \( g \) is the acceleration due to gravity on Earth's surface and \( R \) is the radius of the Earth.

Step 2: Set up the equation

We need to find the height \( h \) where the gravity becomes \( \frac{g}{9} \). Set \( g' = \frac{g}{9} \). This gives us:\[ \frac{g R^2}{(R + h)^2} = \frac{g}{9} \] Simplifying gives:\[ \frac{R^2}{(R + h)^2} = \frac{1}{9} \]

Step 3: Solve for \( R + h \)

Take the square root of both sides to solve for \( R + h \):\[ \frac{R}{R + h} = \frac{1}{3} \] Cross-multiply to obtain:\[ 3R = R + h \]

Step 4: Isolate \( h \)

Rearrange the equation to isolate \( h \):\[ 3R - R = h \] Which simplifies to:\[ h = 2R \]

Step 5: Choose the correct answer

From the simplification, we found \( h = 2R \). Looking at the options, the correct answer is (d) \( 2R \).

Key Concepts

Gravitational Force

Gravitational force is an attractive force that acts between two masses. It is the force that pulls objects towards the Earth, and it is essential to understand when we study gravitational acceleration.

• Newton's Universal Law of Gravitation states that the force between any two masses is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• This can be expressed with the equation: \( F = G \frac{m_1 m_2}{r^2} \) where \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses, and \( r \) is the distance between the centers of these masses.

For objects close to the Earth's surface, this force is usually described by the acceleration due to gravity, denoted as \( g \). This is approximately \( 9.8 \, \text{m/s}^2 \).
Understanding gravitational force allows us to explore interesting phenomena like how gravity decreases with height, as described in the original exercise.

Radius of the Earth

The Earth's radius is a fundamental factor in calculations related to gravity. It refers to the distance from the Earth's center to its surface, typically given as approximately \( 6,371 \, \text{km} \).

• Knowing the Earth's radius helps us understand how gravitational force diminishes as we move away from the Earth's surface.
• This radius plays a crucial role in the equation used to describe gravitational acceleration at a particular height above the Earth's surface, like in the formula:\( g' = \frac{g R^2}{(R + h)^2} \)\.

In this formula, \( R \) is the Earth's radius, \( h \) is the height above the surface, and \( g' \) is the decreased gravitational acceleration at that height.
The radius of the Earth provides a reference point in computations used in a variety of physics problems, particularly those dealing with gravitational variations with height.

Height and Gravity Relationship

The relationship between height and gravity is crucial for understanding gravitational acceleration variations above the Earth's surface. As we move away from the Earth, gravity decreases. This phenomenon is due to the inverse square law of distance mentioned earlier.

• The formula \( g' = \frac{g R^2}{(R + h)^2} \)\ indicates that as height \( h \) increases, the gravitational acceleration \( g' \)\ decreases.
• The decrease in gravitational force with increasing height is why astronauts experience microgravity in space.

In the original exercise, solving the equation \( \frac{R^2}{(R + h)^2} = \frac{1}{9} \)\ shows the height at which gravity decreases to a ninth of its value as \( h = 2R \)\.
This calculation reveals the significant effect of height on gravitational force and demonstrates how mathematical equations can model such physical phenomena.