Problem 725 - JEE Physics

Question

A body of mass \(\mathrm{m} \mathrm{kg}\) starts falling from a point \(2 \mathrm{R}\) above the earth's surface. Its \(\mathrm{K} . \mathrm{E}\). when it has fallen to a point ' \(\mathrm{R}\) ' above the Earth's surface \(=\ldots \ldots \ldots \ldots J\) [R - Radius of Earth, M-mass of Earth G-Gravitational constant \(]\) (A) \((1 / 2)[(\mathrm{GMm}) / \mathrm{R}]\) (B) \((1 / 6)[(\mathrm{GMm}) / \mathrm{R}]\) (C) \((2 / 3)[(\mathrm{GMm}) / \mathrm{R}]\) (D) \((1 / 3)[(\mathrm{GMm}) / \mathrm{R}]\)

Step-by-Step Solution

Answer

The short answer to the problem is: The kinetic energy (KE) of the object when it has fallen to a point R above the Earth's surface is given by option (B) \(\frac{1}{6}(\frac{GMm}{R})\).

Step 1: Identify the Initial and Final Points

We need to first identify the initial and final positions of the object. Initially, the object is at a height of 2R above the Earth's surface, and finally, it falls to a height of R above the Earth's surface.

Step 2: Conservation of Mechanical Energy

According to the conservation of mechanical energy, the total mechanical energy (kinetic energy + gravitational potential energy) remains constant. In other words, the initial total mechanical energy equals the final total mechanical energy: Initial total mechanical energy = Final total mechanical energy

Step 3: Calculate the Initial and Final Gravitational Potential Energies

We need to find the gravitational potential energy (GPE) at both initial and final positions. Since GPE is given by the formula: GPE = \(-\frac{GMm}{r}\) Initial GPE = \(-\frac{GMm}{3R}\) Final GPE = \(-\frac{GMm}{2R}\)

Step 4: Calculate the Final Kinetic Energy (KE)

Since the body is initially at rest, its initial KE is equal to 0. Now we need to calculate the final kinetic energy when the fall reaches a height of R above the Earth's surface. We start by applying the conservation of mechanical energy: Initial KE + Initial GPE = Final KE + Final GPE 0 + \(-\frac{GMm}{3R}\) = Final KE + \(-\frac{GMm}{2R}\)

Step 5: Solve for the Final Kinetic Energy (KE)

We can now solve for the final kinetic energy: Final KE = \(\frac{GMm}{2R} - \frac{GMm}{3R}\) Final KE = \(\frac{GMm}{6R}\) Thus, the correct answer is (B) \(\frac{1}{6}(\frac{GMm}{R})\).

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy possessed by an object due to its position within a gravitational field. It is a crucial concept in understanding the energy transfers involved when objects move through gravitational fields, such as when they are dropped or rise. Gravitational potential energy is given by the formula: \[ \text{GPE} = -\frac{GMm}{r}\]where:

• \(G\) is the gravitational constant,
• \(M\) is the mass of the Earth,
• \(m\) is the mass of the object, and
• \(r\) is the distance from the center of the Earth to the object.

This formula indicates that gravitational potential energy is negative, showing the work done against gravitational forces to bring the object from infinity to a distance \(r\). Gravitational potential energy decreases as an object falls towards the Earth’s surface, which then transforms into kinetic energy. Understanding this transformation is key when solving problems involving energy in gravitational fields.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only converted from one form to another. In our context involving a falling object, this means that the total mechanical energy (sum of kinetic and gravitational potential energy) of the object remains constant throughout its fall. When an object falls, its gravitational potential energy decreases and gets converted into kinetic energy, causing the object to move faster. We equate the initial total mechanical energy with the final total mechanical energy:

• Initial total mechanical energy: Initial KE + Initial GPE
• Final total mechanical energy: Final KE + Final GPE

By solving this equation, we determine the changes in energy as an object transitions from one height to another. This principle is foundational in physics as it allows us to solve for variables when dealing with energy transformation problems, such as determining the kinetic energy of a falling object at different points in its path.

Mechanical Energy

Mechanical energy is the sum of kinetic energy and potential energy in a system. It provides a comprehensive view of the energy in motion and stored energy due to the system’s configuration or position. In the problem involving the object's fall, the system's mechanical energy initially includes only gravitational potential energy because the object starts from rest. As the object falls, kinetic energy increases while gravitational potential energy decreases. However, the sum of kinetic energy and potential energy remains unchanged due to the conservation of energy. Mechanical energy can be visualized in systems where forces cause motion or deformation. A deep understanding of mechanical energy allows us to make predictions about the motion and energy distribution in various scenarios, making it a vital concept in physics and engineering. It sheds light on the intricate dance between movement (kinetic energy) and position (potential energy) in dynamic systems.