H C Verma solutions, Introduction to Physics, Objective-I, Chapter-1, Concepts of Physics, Part-I Introduction to Physics

HC Verma Solutions – Concepts of Physics Part 1

Chapter 1: Introduction to Physics – Objective I

Q1. Which of the following sets cannot enter into the list of fundamental quantities in any system of units?

• (a) length, mass, and velocity
• (b) length, time and velocity
• (c) mass, time and velocity
• (d) length, time and mass

Answer: (b)

Solution: Fundamental quantities must be independent of each other. In option (b), velocity depends on length and time (v = L/T), so it cannot be chosen as a fundamental quantity. Also, without mass the system is incomplete. Hence option (b) is invalid.

Q2. A physical quantity is written as n·u, where u is the unit and n is the numerical value. If the same quantity is expressed in different units, then

• (a) n ∝ u
• (b) n ∝ u²
• (c) n ∝ √u
• (d) n ∝ 1/u

Answer: (d)

Solution: For any physical quantity, the product of the numerical value and the unit remains constant: n × u = constant. If the unit chosen is larger, the numerical value becomes smaller. Hence, n ∝ 1/u.

Q3. Suppose a quantity x has dimensions [x] = M^aL^bT^c. Then the physical quantity mass

• (a) can always be expressed in terms of L, T, and x
• (b) can never be expressed in terms of L, T, and x
• (c) may be expressed if a = 0
• (d) may be expressed if a ≠ 0

Answer: (d)

Solution: If a = 0, then x has no dependence on mass, so mass cannot be expressed. If a ≠ 0, mass is already included in the dimensional formula of x, so it can be expressed in terms of L, T, and x.

Q4. A dimensionless quantity

• (a) never has a unit
• (b) always has a unit
• (c) may or may not have a unit
• (d) does not exist

Answer: (c)

Solution: Some dimensionless quantities like angles have units (radian), while others such as refractive index are pure ratios without units. Hence, a dimensionless quantity may or may not have a unit.

Q5. A unitless quantity

• (a) never has a nonzero dimension
• (b) always has a nonzero dimension
• (c) may have a nonzero dimension
• (d) does not exist

Answer: (a)

Solution: If a quantity has no unit, it must also be dimensionless. Therefore, it can never have a nonzero dimension.

Q6. Evaluate using dimensional analysis:

∫ dx / √(2ax - x²) = aⁿ sin⁻¹(x/a - 1)

• (a) 0
• (b) -1
• (c) 1
• (d) none of these

Answer: (a)

Solution: The term inside sin⁻¹ must be dimensionless, so x and a have the same dimension (length). The left-hand side has dimensions [L]/[L] = [1] (dimensionless). The right-hand side is [aⁿ] × [dimensionless] = [Lⁿ]. For equality, n = 0. Thus, the answer is (a).