91Ó°ÊÓ

For a two-body system in absence of external forces, the kinetic energy as measured from ground frame is \(K_{0}\) and from centre of mass frame is \(K_{\mathrm{cm}}\). Pick up the correct statements. (1) The kinetic energy as measured from centre of mass frame is least. (2) Only the portion of energy \(K_{\mathrm{cm}}\) can be transformed from one form to another due to internal changes in the system. (3) The system always retains at least \(K_{0}-K_{\mathrm{cm}}\) amount of kinetic energy as measured from ground frame irrespective of any kind of internal changes in the system. (4) The system always retains at least \(K_{\mathrm{cm}}\) amount of kinetie energy as measured from ground frame irrespection energy as measured from ground frame irrespective of any kind of internal changes in the system

Step by step solution

01

Understanding Systems of Reference

In a two-body system without external forces, two reference frames are considered: the ground frame and the center of mass frame. The kinetic energy from the ground frame is denoted as \( K_0 \), and from the center of mass frame is \( K_{\mathrm{cm}} \).

02

Comparing Kinetic Energies

In the center of mass (CM) frame, only the relative motion of the bodies contributes to kinetic energy, so \( K_{\mathrm{cm}} \leq K_0 \). This inequality implies the kinetic energy is least in the center of mass frame, supporting statement (1).

03

Energy Transformation Potential

The energy \( K_{\mathrm{cm}} \) is due to the relative motion of the bodies and can be transformed into other forms of energy through internal processes. However, energy \( K_{0} - K_{\mathrm{cm}} \), due to the motion of the center of mass, remains unchanged regardless of internal changes, affirming statement (2).

04

Minimum Retained Kinetic Energy in Ground Frame

Since \( K_{0} = K_{\mathrm{cm}} + \) energy due to CM motion, \( K_{0} - K_{\mathrm{cm}} \) is constant and signifies the minimum energy retained in the ground frame amidst internal changes, justifying statement (3).

05

Validity of Remaining Statement

Statement (4) implies \( K_{\mathrm{cm}} \) is always retained in the ground frame irrespective of internal changes. However, internal changes can convert \( K_{\mathrm{cm}} \) to different forms of energy, so strictly retaining \( K_{\mathrm{cm}} \) isn't guaranteed. Statement (4) is incorrect.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Center of Mass Frame

The center of mass frame is a reference frame in which the total momentum of a system is zero. This means that the center of mass (CM) itself is at rest or moving at a constant velocity. In this frame, kinetic energy is due purely to the relative motion of the bodies within the system. Compared to the ground frame, the kinetic energy in the CM frame, denoted as \( K_{\mathrm{cm}} \), is always less than or equal to the total kinetic energy observed in the ground frame. Since the CM frame isolates the internal interactions, it provides a simpler perspective for analyzing the system's dynamics.

Ground Frame

The ground frame, by contrast, is an inertial frame of reference typically considered fixed relative to the Earth's surface. In this frame, known as the lab or observer's frame, kinetic energy is measured by observing all movements, including the motion of the center of mass itself. The total kinetic energy in this frame is represented as \( K_0 \). This quantity is always larger or equal to the kinetic energy in the center of mass frame due to the inclusion of the CM movement. It serves as a basic and practical reference for everyday observations and analyses.

Energy Transformation

Energy transformation refers to the conversion of energy from one form to another. In a two-body system, energy can transform internally while the total energy remains conserved. The energy due to relative motion, \( K_{\mathrm{cm}} \), is particularly versatile: it can shift from kinetic to potential or other energy types due to internal interactions, like springs or gravitational forces. This transformation doesn't affect the motion of the center of mass, so \( K_0 - K_{\mathrm{cm}} \) remains unchanged, reflecting the conserved energy tied to the CM's motion.

Two-Body System

A two-body system consists of two interacting objects or point masses considered together. When analyzing such a system, especially without external forces, the center of mass frame and ground frame become pivotal. The internal forces between the two masses can lead to changes in \( K_{\mathrm{cm}} \), allowing observations of system dynamics and energy exchanges. However, \( K_0 - K_{\mathrm{cm}} \) ensures a consistent measure of the system's kinetic energy due to center of mass motion, which is immune to internal changes, showcasing how the system balances and transforms energy in isolated situations.