Problem 728 - JEE Physics

Question

Energy required to move a body of mass \(\mathrm{m}\) from from an orbit of radius \(2 \mathrm{R}\) to \(3 \mathrm{R}\) is \(\ldots \ldots \ldots \ldots\) (A) \(\left[(\mathrm{GMm}) /\left(12 \mathrm{R}^{2}\right)\right]\) (B) \(\left[(\mathrm{GMm}) /\left(3 \mathrm{R}^{2}\right)\right]\) (C) \([(\mathrm{GMm}) /(8 \mathrm{R})]\) (D) \([(\mathrm{GMm}) /(6 \mathrm{R})]\)

Step-by-Step Solution

Answer

The energy required to move a body of mass m from an orbit of radius 2R to 3R is \(\Delta U = \frac{GMm}{6R}\). The correct answer is (D).

Step 1: Calculate Initial Potential Energy

In the initial state, the body is in an orbit of radius 2R. The gravitational potential energy at this radius can be calculated using the formula: \(U_1 = -\frac{GMm}{r_1}\) Substituting the given radius, \(r_1 = 2R\): \(U_1 = -\frac{GMm}{2R}\)

Step 2: Calculate Final Potential Energy

In the final state, the body is in an orbit of radius 3R. The gravitational potential energy at this radius can be calculated using the same formula: \(U_2 = -\frac{GMm}{r_2}\) Substituting the given radius, \(r_2 = 3R\): \(U_2 = -\frac{GMm}{3R}\)

Step 3: Find the Difference in Potential Energy

The energy required to move the body from radius 2R to 3R is the difference in potential energy between the two orbits: \(\Delta U = U_2 - U_1\) Substituting the values we calculated for \(U_1\) and \(U_2\): \(\Delta U = -\frac{GMm}{3R} - \left(-\frac{GMm}{2R}\right)\)

Step 4: Simplify the Expression

Now we simplify the expression for the change in potential energy: \(\Delta U = \frac{GMm}{2R} - \frac{GMm}{3R} = \frac{3GMm - 2GMm}{6R} = \frac{GMm}{6R}\)

Step 5: Match the Answer with the Given Options

Matching the expression we found for the change in potential energy with the answer options given in the exercise, we find that the correct answer is: \(\Delta U = [(GMm) /(6R)]\) So, the correct answer is (D).

Key Concepts

Orbital Mechanics

Orbital mechanics is a fundamental element of understanding how objects move in space under the influence of gravity. When we talk about orbits, we often deal with celestial bodies moving in predictable paths around larger masses, like planets orbiting a star or moons orbiting a planet. This movement is governed by the gravitational force acting between the bodies.

In our example exercise, a body is moving between two circular orbits with different radii: from a closer orbit at a radius of \(2R\) to a further orbital path at \(3R\). The energy transition between these orbits involves understanding potential energy at different distances. The further away an object is from the massive body it orbits, the less negative its gravitational potential energy, meaning it has a higher potential.

Knowledge of orbital mechanics is crucial because it helps scientists calculate spacecraft trajectories, predict satellite positions, and design stable orbits for satellites to maintain communication networks or provide GPS data. Understanding these principles is essential for missions involving travel between celestial bodies in our solar system.

Energy Conservation

Energy conservation is a key concept in physics, explaining that energy cannot be created or destroyed, only transformed from one form to another. In the context of gravitational potential energy and orbital mechanics, this means that the total energy of a system remains constant unless external forces do work on it.

In the problem given, we started by calculating the initial gravitational potential energy when the body is at a radius of \(2R\) and then found the energy at a radius of \(3R\). The change in gravitational potential energy reflects the work done to move the object between these two orbital distances.

• The initial potential energy was \(-\frac{GMm}{2R}\).
• The final potential energy was \(-\frac{GMm}{3R}\).
• The energy change is simply the difference between these two states, which resulted in a calculated result of \(\frac{GMm}{6R}\).

This calculation demonstrates energy conservation in action. As the body moves into a higher orbit, gravitational potential energy increases, requiring an input of energy to make this transition. It's crucial for all physics problems to remember that all forms of energy transformations adhere to the principle of conservation.

Physics Problems

Solving physics problems often requires a systematic approach where principles are broken down into actionable steps. Key physics concepts help us to understand and predict natural phenomena and are pivotal in solving exercises such as calculating gravitational potential energy changes in orbital mechanics.

Firstly, we need to recognize how each of the given elements, such as mass \(m\), gravitational constant \(G\), and radii \(R\), relate within the context of gravitational forces. Knowing how to apply the formula \(U = -\frac{GMm}{r}\) to find potential energy at specific points in an orbit helps in understanding how much energy is involved in changing orbits.

Next, simplifying equations is crucial when working through physics problems. Breaking down the problem step by step as in our provided solution allows you to follow the logical steps needed to find the solution. This helps not only in arriving at the correct answer but also in understanding how each part of the physics problem fits together and applies to real-world scenarios.