Problem 760 - JEE Physics

Question

A shell of mass \(\mathrm{M}\) and radius \(\mathrm{R}\) has point mass \(\mathrm{m}\) placed at a distance \(r\) from its center. The gravitational potential energy \(\mathrm{U}(\mathrm{r})-\mathrm{v}\) will be

Step-by-Step Solution

Answer

The gravitational potential energy \(U(r)\) of a point mass \(m\) located at a distance \(r\) from the center of a shell of mass \(M\) and radius \(R\) is given by the formula: \(U(r) = -\dfrac{G m M}{r}\), where G is the gravitational constant.

Step 1: Write down the formula for potential energy

To calculate the gravitational potential energy, we need to use the formula: \(U(r) = -\dfrac{G m M}{r}\) Where U(r) is the potential energy as a function of r, G is the gravitational constant (approximately \(6.674 \times 10^{-11}\) N(m/kg)²), m is the given point mass, M is the mass of the shell, and r is the distance from the center of the shell.

Step 2: Compute U(r)-v

The exercise asks to compute \(U(r)-v\). But there isn't information about the value of "v" anywhere in the exercise. We'll assume that this is an error and that the desired expression is only U(r). So, we need to find the gravitational potential energy at distance r from the center of the shell. By substituting the given values into the potential energy formula, we get: \(U(r) = -\dfrac{G m M}{r}\) Which is the potential energy as a function of the distance from the center of the shell.

Key Concepts

Newton's Law of Gravitation

Newton's Law of Gravitation is a fundamental principle that describes how every mass attracts every other mass in the universe. It's the force that we commonly refer to as gravity, and it's extremely important for understanding how objects interact in space.
The law states that the gravitational force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. Mathematically, this is expressed as:

• \( F = \dfrac{G \cdot m_1 \cdot m_2}{r^2} \)
  • \( F \) is the force of gravity
  • \( m_1 \) and \( m_2 \) are the masses of the two objects
  • \( r \) is the distance between the centers of the two masses
  • \( G \) is the gravitational constant

This formula is vital for calculating how much one object will attract another, and it applies universally, from apples falling from trees to the orbits of planets around the sun.

Gravitational Constant

The gravitational constant, often denoted as \( G \), is a key part of Newton's Law of Gravitation. It helps determine the gravitational force between two masses. \( G \) is a universal constant, meaning it's a fixed value that doesn't change regardless of where you are in the universe.
The approximate value of this constant is \( 6.674 \times 10^{-11} \) N(m/kg)². This small number is why gravity feels weak compared to other forces like electromagnetism; the force of gravity decreases very rapidly with distance.
In formulas like the gravitational potential energy formula, \( G \) is used to calculate how much potential energy is stored in the position of a mass in a gravitational field. For example, if you fill in the values for \( m \), \( M \), and \( r \), you can determine the gravitational potential energy \( U(r) = -\dfrac{G \cdot m \cdot M}{r} \), illustrating how \( G \) directly influences the energy between two masses.

Point Mass

Point mass is a theoretical term used in physics to simplify the study of forces like gravity. It's an idealized point where the entire mass of an object is assumed to be concentrated at a single point.
When we calculate gravitational interactions, simplifying a complex object as a point mass makes it easier to apply Newton's Law of Gravitation. This is because the law assumes that both interacting bodies are point masses, ensuring that their actual shape or size doesn't complicate the calculations.
For example, in the problem given, we deal with a point mass \( m \), which greatly simplifies the calculation of its gravitational interaction with the shell. By assuming everything is concentrated at one point, you can use the formula for potential energy, \( U(r) = -\dfrac{G \cdot m \cdot M}{r} \), straightforwardly without worrying about distribution of mass in the shell.