Question

STATEMENT-1 If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant. because STATEMENT-2 The linear momentum of an isolated system remains constant. (A) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1 (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement- 2 is False (D) Statement-1 is False, Statement-2 is True

Short answer

(A) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1

Step-by-step solution

Understanding Statement-1

First, analyze Statement-1: 'If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant.' This is a principle derived from Newton's first law of motion, which states that an object will remain at rest or in uniform motion unless acted upon by an external force. In this context, the absence of external torque implies that there is no external force causing rotational acceleration, which in turn means that there will be no change in the linear velocity of the center of mass.

Understanding Statement-2

Examine Statement-2: 'The linear momentum of an isolated system remains constant.' This statement is based on the conservation of momentum principle which states that if no external forces (and hence no external torques) are acting on a system, the total linear momentum of the system remains constant over time.

Analyzing the Relationship between the Statements

Determine the relationship between the two statements: Considering the principles of mechanics, Statement-1 is a consequence of Statement-2. The constant velocity of the center of mass implies that the momentum (which is the product of mass and velocity) is also constant. Since Statement-2 is a general statement about the conservation of momentum, which implies a constant velocity in the absence of external forces, it is a valid explanation for why the velocity of the center of mass remains constant as per Statement-1.

Selecting the Correct Option

Based on the analysis of both statements and their relationship, we can conclude that both Statement-1 and Statement-2 are true, and Statement-2 is indeed the correct explanation for Statement-1. Therefore, the correct option is (A).

Key concepts

External Torque

External torque is a concept that matters greatly when analyzing the motion of bodies, particularly in rotational dynamics. Torque can be thought of as the rotational equivalent of force; it is what causes an object to rotate. When we refer to an external torque, we're talking about a torque that comes from outside the system in question, altering the object's angular velocity.

Imagine spinning a bike wheel; your hand exerts an external torque on the wheel. If no external torque is applied, such as when the wheel is spinning freely, its angular velocity remains consistent. This ties back into the exercise statement, showing how the absence of external torque ensures the constancy of the center of mass velocity.

Center of Mass

The center of mass is a point in a body or system of bodies which moves as if all of the mass were concentrated at that point and all external forces were applied there. It is an incredibly useful concept because it simplifies the analysis of motion.

Whether we're looking at a single object or a complex system, calculating the center of mass gives us a simplified way to predict the motion of the system. In the problem we're examining, understanding that the velocity of the center of mass will not change without an external force or torque is key to understanding why the body's overall motion remains constant.

Newton's First Law of Motion

Newton's first law of motion, often referred to as the law of inertia, states that an object at rest will stay at rest, and an object in motion will stay in motion at a constant velocity, unless acted upon by a net external force.

This foundational principle explains why the center of mass maintains a consistent velocity in the absence of external torque—because torque is a type of force that can change the state of motion. Applying this law helps us understand the broader implications of the way objects behave when left alone and how forces change those behaviors.

Isolated System

An isolated system is a physical system that does not interact with its surroundings. This means no matter or energy is exchanged with the environment. Isolated systems are idealizations, but they are very useful for understanding physical processes like the conservation of momentum.

In the context of our textbook problem, considering the system as isolated allows us to apply conservation laws without worrying about external influences, ensuring the predictability of the system's behavior over time.

Linear Momentum

Linear momentum is a measure of an object's motion, calculated as the product of its mass and velocity. It's a vector quantity, which implies it has both magnitude and direction. A pivotal concept in physics, conservation of linear momentum, tells us that in the absence of external forces, the total linear momentum of a system remains unchanged.

This principle is directly related to both the problem statement and Newton's first law of motion. It indicates that without external influences, the momentum of the center of mass, and hence its velocity, will not alter. This understanding aids students immensely in both predicting and analyzing the motion of objects and systems.