Questions OBJECTIVE - I and Answer (Introduction to Physics) HC Verma Part 1

 Q#1

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). 
Two qualities of fundamental quantities are,
(a) The fundamental quantities should be independent of each other. In (b) velocity is dependent on length and time.

(b) All other quantities may be expressed in terms of the fundamental quantities. But in (b), the mass of an object can not be expressed in these terms.       

Q#2
A physical quantity is measured and the result expressed as nu where u is the unit used and n is the numerical value. If the result is expressed in various units then
(a) n ∝ size of u
(b) n ∝ u²
(c) n ∝ √u 
(d) n ∝ 1/u.

Answer: (d). 
Bigger the unit smaller will be the numerical value n. So n is inversely proportional to the size of the unit. i.e. n x u = constant. For example, a distance of 5 km can be expressed as 5000 m, 500000 cm, 5000000 mm.

5000 m = 5000(1/1000) km = 5 km
500000 cm = 500000(1/100000) km = 5 km
5000000 mm = 5000000(1/1000000) km = 5 km.      

 Q#3
Suppose a quantity x can be dimensionally represented in terms of M, L and T, that is [x] = M^aL^bT^c. The quantity mass 
(a) can always be dimensionally represented in terms of L, T, and x,
(b) can never be dimensionally represented in terms of L, T, and x,
(c) maybe represented in terms of L, T, and x if a = 0.
(d) maybe represented in terms of L, T, and x if a ≠ 0.
      
Answer: (d). 
If a = 0, then the quantity has no mass component, hence it cannot be represented in terms of L, T, and x. If a ≠ 0, then it has a mass component and it can be expressed in terms of L, T, and x. So, the option (d) is true.   

Q#4
A dimensionless quantity
(a) never has a unit,
(b) always has a unit,
(c) may have a unit
(d) does not exist.  

Answer: (c). 
A dimensionless quantity may or may not have a unit. For example, the angle has a unit as radian, but it is a dimensionless quantity while specific gravity is also dimensionless but unitless. Hence the option (c).      

Q#5
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). 
A unitless quantity is also dimensionless. Hence it can never have a nonzero dimension.     

Q#6
∫  dx/√(2ax – x²) = aⁿ sin⁻¹[x/a – 1].
The value of n is
(a) 0,
(b) –1,
(c) 1,
(d) non of these.
You may use dimension analysis to solve the problem.

Answer: (a). 
We assume x as length, hence x has dimensions = [L]. The whole term Sin⁻¹(x/a – 1) is an angle (radian) hence it is dimensionless. In this (x/a – 1) is the ratio (Sine) hence dimensionless, thus x and a must-have the same dimension that of L. Now,

[∫ dx/√(2ax – x²)] = [L/√L²] = [L⁰]  and
[aⁿ Sin⁻¹(x/a – 1)] = [Lⁿ]

Equating the dimensions of both sides,

[L⁰] = [Lⁿ]
n = 0.

Hence the option (a).