Problem 28 - Master Resource Book in JEE Main Physics

Question

How much energy will be necessary for making a body of \(500 \mathrm{~kg}\) escape from the earth? \(\left(g=9.8 \mathrm{~ms}^{-2}\right.\), radius of earth \(\left.=6.4 \times 10^{6} \mathrm{~m}\right)\) (a) About \(9.8 \times 10^{6} \mathrm{~J}\) (b) About \(6.4 \times 10^{8} \mathrm{~J}\) (c) About \(3.1 \times 10^{10} \mathrm{~J}\) (d) About \(27.4 \times 10^{12} \mathrm{~J}\)

Step-by-Step Solution

Answer

Option (c) \( 3.1 \times 10^{10} \text{ J} \) is the correct answer.

Step 1: Understand the Problem

We need to determine the necessary energy to make a 500 kg body escape from Earth's gravitational field. This requires calculating the escape energy using the body's mass, Earth's gravitational constant, and Earth's radius.

Step 2: Recall the Formula for Escape Energy

The escape energy can be calculated using the formula for gravitational potential energy required to escape the Earth's gravitational pull: \[ E = \frac{G M_e m}{R_e} \] where:- \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \mathrm{Nm}^2/ ext{kg}^2 \),- \( M_e \) is the mass of the Earth \( 5.972 \times 10^{24} \) kg,- \( m \) is the mass of the object (500 kg),- \( R_e \) is the radius of the Earth \(6.4 \times 10^6 \text{ m}\).

Step 3: Calculate the Gravitational Force Constant

First, calculate the magnitude of Earth's gravity using the formula:\[ g = \frac{G M_e}{R_e^2} \]Substitute \( G = 6.674 \times 10^{-11} \mathrm{Nm}^2/ ext{kg}^2 \), \( M_e = 5.972 \times 10^{24} \) kg, and \( R_e = 6.4 \times 10^6 \text{ m}\).Verify \( g \approx 9.8 \mathrm{m/s}^2\). This confirms the formulas and given values align.

Step 4: Calculate Escape Energy Directly

Using the verified values, substitute into the formula for escape energy:\[ E = \frac{6.674 \times 10^{-11} \times 5.972 \times 10^{24} \times 500}{6.4 \times 10^6} \]Calculate this expression to find the total escape energy required for the 500 kg object.

Step 5: Simplify and Solve for Energy

Perform the necessary calculations:\[ E \approx \frac{1.9898 \times 10^{14}}{6.4 \times 10^6} \]\[ E \approx 3.1 \times 10^{10} \text{ J} \]Therefore, the required energy is approximately \( 3.1 \times 10^{10} \text{ J} \).

Step 6: Compare and Choose the Closest Option

Compare the computed energy \( 3.1 \times 10^{10} \text{ J} \) to the given choices. The correct choice is clearly option (c) \( 3.1 \times 10^{10} \text{ J} \).

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy possessed by an object due to its position relative to Earth. In the context of escape velocity, it refers to the work needed to move an object from Earth's surface to a point far away where Earth's gravitational pull doesn't impact it significantly.

Gravitational potential energy depends on mass, gravity, and height from Earth. The formula is: \[ E = \frac{G M_e m}{R_e} \] where:

• \(E\) is the gravitational potential energy,
• \(G\) is the gravitational constant,
• \(M_e\) is Earth's mass,
• \(m\) is the object's mass,
• \(R_e\) is Earth's radius.

To achieve escape velocity, this potential energy matches the energy needed to overcome Earth's gravitational pull. Understanding this concept helps in calculating how much energy is required to send a spacecraft or satellite beyond Earth's gravitational influence.

Gravitational Constant

The gravitational constant \( G \) is a key figure in physics, acting as a universal constant that affects all gravitational interactions. It's a measure of the strength of gravity in the universe, appearing in Isaac Newton's law of universal gravitation.

This constant has a value of \( 6.674 \times 10^{-11} \mathrm{m}^3/kg/s^2 \). When calculating escape velocity, \( G \) helps us determine how strong Earth's pull is on a given mass.

Due to its universal nature, \( G \) ensures that calculations involving gravity are consistent, making it crucial for scientists studying astrophysics and cosmology. It's used to calculate forces not only on Earth but also between cosmic entities like planets, stars, and galaxies.

Radius of Earth

The radius of Earth \( R_e \) is vital in computations related to gravity and escape energy. It's essentially the distance from Earth's center to its surface. For calculations, we use a standard radius of \( 6.4 \times 10^6 \text{ m} \).

This value helps us calculate the gravitational pull and the required velocity for an object to break free from Earth's atmosphere without returning.

In context, the radius directly affects escape velocity: the larger the planet, the stronger the gravitational pull due to increased surface area and mass. Hence, knowing Earth's radius allows precise planning for space missions, impacting decisions on launch energy requirements and trajectories.

Mass of Earth

The mass of Earth \( M_e \) is a fundamental parameter used in gravitational calculations. Earth’s mass is measured as \( 5.972 \times 10^{24} \text{ kg} \). This immense quantity comes into play when determining how much force Earth exerts on objects within its gravitational field.

Earth's mass directly influences the gravitational potential energy calculation for an object on its surface or nearby. It appears in both the numerator of the gravitational potential energy formula and in the formula for calculating gravitational force:

\[ g = \frac{G M_e}{R_e^2} \]

In essence, the mass of Earth dictates the intensity of gravity we experience, impacting everything from tides to the trajectory of spaceships attempting to leave its sphere of influence. Understanding Earth's mass is crucial for accurate, predictive models in physics and astronomy.