Class 11 NCERT PHYSICS • Chapter 3: Motion in a Plane • Question 21

QUESTION 21

Read each statement below carefully and state, with reasons and examples, if it is true or false :

A scalar quantity is one that

(a) is conserved in a process

(b) can never take negative values

(c) must be dimensionless

(d) does not vary from one point to another in space

(e) has the same value for observers with different orientations of axes.

SOLUTION

We need to evaluate five statements about scalar quantities to determine their validity based on physical definitions and examples.

Concept: Scalar

A scalar is a physical quantity that is completely specified by its magnitude and unit. Mathematically, a scalar is defined by its invariance under the rotation of the coordinate system, meaning its value does not depend on the observer's orientation.

Given:

The following statements regarding a scalar quantity:

(a) It is conserved in a process.
(b) It can never take negative values.
(c) It must be dimensionless.
(d) It does not vary from one point to another in space.
(e) It has the same value for observers with different orientations of axes.

Solving:

Analysis of Statement (a):
- This statement is False. Being a scalar does not guarantee that the quantity is conserved. Conservation depends on the specific physical laws governing a process.
- Example: Kinetic energy is a scalar quantity, but it is not conserved in an inelastic collision, where it is partially converted into internal energy or heat.
Analysis of Statement (b):
- This statement is False. Scalars can take negative values depending on the choice of reference or the nature of the quantity.
- Example: Temperature can be negative (e.g., $-15\,^{\circ}\text{C}$ ). Similarly, work done or potential energy can be negative.
Analysis of Statement (c):
- This statement is False. Most physical scalars have dimensions and units.
- Example: Mass is a scalar with the dimension $[M]$ , and time is a scalar with the dimension $[T]$ . Only specific ratios, like relative density, are dimensionless scalars.
Analysis of Statement (d):
- This statement is False. A scalar can change its value from one point to another, forming what is known as a scalar field.
- Example: Atmospheric pressure or the temperature in a room varies from one spatial coordinate $(x, y, z)$ to another.
Analysis of Statement (e):
- This statement is True. This is the defining property of a scalar quantity. While the components of a vector change when the axes are rotated, a scalar remains invariant.
- Example: The mass of an object or the distance between two points remains the same regardless of how the coordinate axes are oriented.

Answer:

(a) False
(b) False
(c) False
(d) False
(e) True