Problem 750 - JEE Physics

Question

A particle of mass \(10 \mathrm{~g}\) is kept on the surface of a uniform sphere of mass \(100 \mathrm{~kg}\) and radius \(10 \mathrm{~cm}\). Find the work to be done against the gravitational force between them to take the particle is away from the sphere \(\left(\mathrm{G}=6.67 \times 10^{-11} \mathrm{SI}\right.\) unit \()\) (A) \(6.67 \times 10^{-9} \mathrm{~J}\) (B) \(6.67 \times 10^{-10} \mathrm{~J}\) (C) \(13.34 \times 10^{-10} \mathrm{~J}\) (D) \(3.33 \times 10^{-10} \mathrm{~J}\)

Step-by-Step Solution

Answer

The work to be done against the gravitational force between the particle and the sphere is \(6.67 \times 10^{-10} \mathrm{J}\). The correct answer is (B).

Step 1: Identify the given information

We are given the following: - Mass of particle (m1) = 10 g = 0.01 kg (converted to kilograms) - Mass of sphere (m2) = 100 kg - Radius of sphere (r) = 10 cm = 0.1 m (converted to meters) - Gravitational constant (G) = 6.67 × 10^-11 N m²/kg²

Step 2: Calculate the gravitational force

We will calculate the gravitational force between the particle and the sphere using the formula F = G * (m1 * m2) / r^2: F = (6.67 × 10^-11 N m²/kg²) * (0.01 kg * 100 kg) / (0.1 m)^2

Step 3: Simplify the expression

Next, we will simplify the expression: F = (6.67 × 10^-11 N m²/kg²) * (1 kg) / 0.01 m² F = 6.67 × 10^-9 N

Step 4: Calculate the work done against gravitational force

Since work is done against the force, the work is equal to the force multiplied by the radius of the sphere (the distance over which the force is applied): W = F * r W = (6.67 × 10^-9 N) * (0.1 m)

Step 5: Simplify the expression and find the answer

Now, we will simplify the expression to obtain the answer: W = 6.67 × 10^-10 J So the correct answer is (B) 6.67 × 10^-10 J.

Key Concepts

Work Done by Gravitational Force

Work is defined as the energy required to move an object against a force. In this scenario, we are focused on the gravitational force that acts between a tiny particle and a large spherical mass. The work done against gravitational force specifically refers to the energy needed to move the particle away from the gravitational pull of the sphere.
To calculate the work done, we first determine the gravitational force between the two masses using the formula:\[ F = \frac{G \, m_1 \, m_2}{r^2} \]This formula helps us find the force in Newtons, where:- \( G \) is the gravitational constant- \( m_1 \) and \( m_2 \) are the masses of the objects- \( r \) is the distance between the centers of the two masses (initially the radius of the sphere, when the particle is on the surface).
Once we have the gravitational force, the work done can be computed as:\[ W = F \, r \]Here, the distance is equivalent to the radius because the particle moves away from the surface of the sphere. By applying this method, you can calculate the energy required to overcome gravitational attraction, which in this exercise is \(6.67 \times 10^{-10} \, \text{J}\).

Gravity and Mass

Gravity is a force of attraction that acts between any two objects with mass. Every bit of matter in the universe exerts a gravitational pull on every other bit. However, the force of gravity becomes significant only when at least one of the objects is very massive, such as a planet or star.
In this exercise, the concept underscores how gravity depends crucially on the mass of the involved objects. The more massive an object, the greater its gravitational pull. So when we talk about the sphere with a mass of 100 kg and a small particle of 10 g, the larger mass noticeably dominates the gravitational force interaction. Still, every bit of mass contributes to the total gravitational force.
Here are the key takeaways about gravity and mass:

• Gravitational force increases with mass—a direct relationship where the force becomes stronger when the masses involved are larger.
• At a constant distance, e.g., the radius of the sphere, the gravitational pull a particle experiences is fixed unless the mass changes.

These insights illustrate how fundamental mass is in determining the exerted gravitational force and how we calculate it based on mass values.

Newton's Law of Universal Gravitation

Sir Isaac Newton formulated the Law of Universal Gravitation in the 17th century, and it remains a cornerstone of physics today. This law states that every point mass attracts every other point mass in the universe 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.
This can be mathematically written as:\[ F = G \frac{m_1 m_2}{r^2} \]Where:- \( F \) is the gravitational force between the masses.- \( G \) is the gravitational constant, \(6.67 \times 10^{-11} \, \text{N} \, \text{m}^2/\text{kg}^2\).- \( m_1 \) and \( m_2 \) are the respective masses of the two objects.- \( r \) is the distance between the centers of the two masses.
This simple yet powerful equation allows us to understand and predict how objects will interact under gravity. Newton's Law helps us to comprehend not only this specific exercise with the sphere and particle, but also the motion of planets, the fall of an apple, and the tides caused by celestial bodies. It reveals the universal nature of gravity across the cosmos.